Mathematics · Textbook solutions

Integrals

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 314 questions

7.2 Integration as an Inverse Process of Differentiation

26 q

Solved Examples

Worked · 4

7.1 Eg.1 $\cos 2x$
7.1 Eg.2 $\int \dfrac{x^3 - 1}{x^2}\,dx$
7.1 Eg.3 $\int (\sin x + \cos x)\,dx$
7.1 Eg.4 $F$

Exercise 7.1

Practice · 22

Ex 7.1 Q1 $\sin 2x$
Ex 7.1 Q2 $\cos 3x$
Ex 7.1 Q3 $e^{2x}$
Ex 7.1 Q4 $(ax + b)^2$
Ex 7.1 Q5 $\sin 2x - 4e^{3x}$
Ex 7.1 Q6 $\int \left(4e^{3x} + 1\right) dx$
Ex 7.1 Q7 $\int x^2 \left(1 - \dfrac{1}{x^2}\right) dx$
Ex 7.1 Q8 $\int \left(ax^2 + bx + c\right) dx$
Ex 7.1 Q9 $\int \left(2x^2 + e^x\right) dx$
Ex 7.1 Q10 $\int \left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)^2 dx$
Ex 7.1 Q11 $\int \dfrac{x^3 + 5x^2 - 4}{x^2}\,dx$
Ex 7.1 Q12 $\int \dfrac{x^3 + 3x + 4}{\sqrt{x}}\,dx$
Ex 7.1 Q13 $\int \dfrac{x^3 - x^2 + x - 1}{x - 1}\,dx$
Ex 7.1 Q14 $\int (1 - x)\sqrt{x}\,dx$
Ex 7.1 Q15 $\int \sqrt{x}\left(3x^2 + 2x + 3\right) dx$
Ex 7.1 Q16 $\int \left(2x - 3\cos x + e^x\right) dx$
Ex 7.1 Q17 $\int \left(2x^2 - 3\sin x + 5\sqrt{x}\right) dx$
Ex 7.1 Q18 $\int \sec x\,(\sec x + \tan x)\,dx$
Ex 7.1 Q19 $\int \dfrac{\sec^2 x}{\csc^2 x}\,dx$
Ex 7.1 Q20 $\int \dfrac{2 - 3\sin x}{\cos^2 x}\,dx$
Ex 7.1 Q21 $\left(\sqrt{x} + \dfrac{1}{\sqrt{x}}\right)$
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Ex 7.1 Q22 $\dfrac{d}{dx}f(x) = 4x^3 - \dfrac{3}{x^4}$
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7.3 Methods of Integration — Substitution

46 q

Solved Examples

Worked · 7

Integrate the following functions w.r.t. $x$ :
7.2 Eg.5(i) $\sin mx$
7.2 Eg.5(ii) $2x \sin (x^2 + 1)$
7.2 Eg.5(iii) $\dfrac{\tan^4 \sqrt{x}\, \sec^2 \sqrt{x}}{\sqrt{x}}$
7.2 Eg.5(iv) $\dfrac{\sin (\tan^{-1} x)}{1 + x^2}$
Find the following integrals:
7.2 Eg.6(i) $\int \sin^3 x \cos^2 x\,dx$
7.2 Eg.6(ii) $\int \dfrac{\sin x}{\sin (x + a)}\,dx$
7.2 Eg.6(iii) $\int \dfrac{1}{1 + \tan x}\,dx$

Exercise 7.2

Practice · 39

Ex 7.2 Q1 $\dfrac{2x}{1 + x^2}$
Ex 7.2 Q2 $\dfrac{(\log x)^2}{x}$
Ex 7.2 Q3 $\dfrac{1}{x + x \log x}$
Ex 7.2 Q4 $\sin x \sin (\cos x)$
Ex 7.2 Q5 $\sin (ax + b) \cos (ax + b)$
Ex 7.2 Q6 $\sqrt{ax + b}$
Ex 7.2 Q7 $x \sqrt{x + 2}$
Ex 7.2 Q8 $x \sqrt{1 + 2x^2}$
Ex 7.2 Q9 $(4x + 2) \sqrt{x^2 + x + 1}$
Ex 7.2 Q10 $\dfrac{1}{x - \sqrt{x}}$
Ex 7.2 Q11 $\dfrac{x}{\sqrt{x + 4}},\ x > 0$
Ex 7.2 Q12 $(x^3 - 1)^{\frac{1}{3}} x^5$
Ex 7.2 Q13 $\dfrac{x^2}{(2 + 3x^3)^3}$
Ex 7.2 Q14 $\dfrac{1}{x (\log x)^m},\ x > 0,\ m \ne 1$
Ex 7.2 Q15 $\dfrac{x}{9 - 4x^2}$
Ex 7.2 Q16 $e^{2x + 3}$
Ex 7.2 Q17 $\dfrac{x}{e^{x^2}}$
Ex 7.2 Q18 $\dfrac{e^{\tan^{-1} x}}{1 + x^2}$
Ex 7.2 Q19 $\dfrac{e^{2x} - 1}{e^{2x} + 1}$
Ex 7.2 Q20 $\dfrac{e^{2x} - e^{-2x}}{e^{2x} + e^{-2x}}$
Ex 7.2 Q21 $\tan^2 (2x - 3)$
Ex 7.2 Q22 $\sec^2 (7 - 4x)$
Ex 7.2 Q23 $\dfrac{\sin^{-1} x}{\sqrt{1 - x^2}}$
Ex 7.2 Q24 $\dfrac{2 \cos x - 3 \sin x}{6 \cos x + 4 \sin x}$
Ex 7.2 Q25 $\dfrac{1}{\cos^2 x (1 - \tan x)^2}$
Ex 7.2 Q26 $\dfrac{\cos \sqrt{x}}{\sqrt{x}}$
Ex 7.2 Q27 $\sqrt{\sin 2x}\, \cos 2x$
Ex 7.2 Q28 $\dfrac{\cos x}{\sqrt{1 + \sin x}}$
Ex 7.2 Q29 $\cot x \log \sin x$
Ex 7.2 Q30 $\dfrac{\sin x}{1 + \cos x}$
Ex 7.2 Q31 $\dfrac{\sin x}{(1 + \cos x)^2}$
Ex 7.2 Q32 $\dfrac{1}{1 + \cot x}$
Ex 7.2 Q33 $\dfrac{1}{1 - \tan x}$
Ex 7.2 Q34 $\dfrac{\sqrt{\tan x}}{\sin x \cos x}$
Ex 7.2 Q35 $\dfrac{(1 + \log x)^2}{x}$
Ex 7.2 Q36 $\dfrac{(x + 1)(x + \log x)^2}{x}$
Ex 7.2 Q37 $\dfrac{x^3 \sin (\tan^{-1} x^4)}{1 + x^8}$
Ex 7.2 Q38 $\displaystyle\int \dfrac{10x^9 + 10^x \log_e 10}{x^{10} + 10^x}\,dx$
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Ex 7.2 Q39 $\displaystyle\int \dfrac{dx}{\sin^2 x \cos^2 x}$
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7.3 Methods of Integration — Trigonometric Identities

27 q

Solved Examples

Worked · 3

Find the following integrals.
7.3 Eg.7(i) $\int \cos^2 x\,dx$
7.3 Eg.7(ii) $\int \sin 2x \cos 3x\,dx$
7.3 Eg.7(iii) $\int \sin^3 x\,dx$

Exercise 7.3

Practice · 24

Ex 7.3 Q1 $\sin^2 (2x + 5)$
Ex 7.3 Q2 $\sin 3x \cos 4x$
Ex 7.3 Q3 $\cos 2x \cos 4x \cos 6x$
Ex 7.3 Q4 $\sin^3 (2x + 1)$
Ex 7.3 Q5 $\sin^3 x \cos^3 x$
Ex 7.3 Q6 $\sin x \sin 2x \sin 3x$
Ex 7.3 Q7 $\sin 4x \sin 8x$
Ex 7.3 Q8 $\dfrac{1 - \cos x}{1 + \cos x}$
Ex 7.3 Q9 $\dfrac{\cos x}{1 + \cos x}$
Ex 7.3 Q10 $\sin^4 x$
Ex 7.3 Q11 $\cos^4 2x$
Ex 7.3 Q12 $\dfrac{\sin^2 x}{1 + \cos x}$
Ex 7.3 Q13 $\dfrac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha}$
Ex 7.3 Q14 $\dfrac{\cos x - \sin x}{1 + \sin 2x}$
Ex 7.3 Q15 $\tan^3 2x \sec 2x$
Ex 7.3 Q16 $\tan^4 x$
Ex 7.3 Q17 $\dfrac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x}$
Ex 7.3 Q18 $\dfrac{\cos 2x + 2\sin^2 x}{\cos^2 x}$
Ex 7.3 Q19 $\dfrac{1}{\sin x \cos^3 x}$
Ex 7.3 Q20 $\dfrac{\cos 2x}{(\cos x + \sin x)^2}$
Ex 7.3 Q21 $\sin^{-1} (\cos x)$
Ex 7.3 Q22 $\dfrac{1}{\cos (x - a) \cos (x - b)}$
Ex 7.3 Q23 $\displaystyle\int \dfrac{\sin^2 x - \cos^2 x}{\sin^2 x \cos^2 x}\,dx$
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Ex 7.3 Q24 $\displaystyle\int \dfrac{e^x (1 + x)}{\cos^2 (e^x x)}\,dx$
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7.4 Integrals of Some Particular Functions

28 q

Solved Examples

Worked · 3

7.4 Eg.8 $\int \frac{dx}{x^2-16}$
7.4 Eg.9 $\int \frac{dx}{x^2-6x+13}$
7.4 Eg.10 $\int \frac{x+2}{2x^2+6x+5}\,dx$

Exercise 7.4

Practice · 25

Ex 7.4 Q1 $\frac{3x^2}{x^6+1}$
Ex 7.4 Q2 $\frac{1}{\sqrt{1+4x^2}}$
Ex 7.4 Q3 $\frac{1}{\sqrt{(2-x)^2+1}}$
Ex 7.4 Q4 $\frac{1}{\sqrt{9-25x^2}}$
Ex 7.4 Q5 $\frac{3x}{1+2x^4}$
Ex 7.4 Q6 $\frac{x^2}{1-x^6}$
Ex 7.4 Q7 $\frac{x-1}{\sqrt{x^2-1}}$
Ex 7.4 Q8 $\frac{x^2}{\sqrt{x^6+a^6}}$
Ex 7.4 Q9 $\frac{\sec^2 x}{\sqrt{\tan^2 x+4}}$
Ex 7.4 Q10 $\frac{1}{\sqrt{x^2+2x+2}}$
Ex 7.4 Q11 $\frac{1}{9x^2+6x+5}$
Ex 7.4 Q12 $\frac{1}{\sqrt{7-6x-x^2}}$
Ex 7.4 Q13 $\frac{1}{\sqrt{(x-1)(x-2)}}$
Ex 7.4 Q14 $\frac{1}{\sqrt{8+3x-x^2}}$
Ex 7.4 Q15 $\frac{1}{\sqrt{(x-a)(x-b)}}$
Ex 7.4 Q16 $\frac{4x+1}{\sqrt{2x^2+x-3}}$
Ex 7.4 Q17 $\frac{x+2}{\sqrt{x^2-1}}$
Ex 7.4 Q18 $\frac{5x-2}{1+2x+3x^2}$
Ex 7.4 Q19 $\frac{6x+7}{\sqrt{(x-5)(x-4)}}$
Ex 7.4 Q20 $\frac{x+2}{\sqrt{4x-x^2}}$
Ex 7.4 Q21 $\frac{x+2}{\sqrt{x^2+2x+3}}$
Ex 7.4 Q22 $\frac{x+3}{x^2-2x-5}$
Ex 7.4 Q23 $\frac{5x+3}{\sqrt{x^2+4x+10}}$
Ex 7.4 Q24 $\int \frac{dx}{x^2+2x+2}$
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Ex 7.4 Q25 $\int \frac{dx}{\sqrt{9x-4x^2}}$
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7.5 Integration by Partial Fractions

29 q

Solved Examples

Worked · 6

7.5 Eg.11 $\int \frac{dx}{(x+1)(x+2)}$
7.5 Eg.12 $\int \frac{x^2+1}{x^2-5x+6}\,dx$
7.5 Eg.13 $\int \frac{3x-2}{(x+1)^2(x+3)}\,dx$
7.5 Eg.14 $\int \frac{x^2}{(x^2+1)(x^2+4)}\,dx$
7.5 Eg.15 $\int \frac{(3\sin\phi-2)\cos\phi}{5-\cos^2\phi-4\sin\phi}\,d\phi$
7.5 Eg.16 $\int \frac{x^2+x+1}{(x+2)(x^2+1)}\,dx$

Exercise 7.5

Practice · 23

Ex 7.5 Q1 $\frac{x}{(x+1)(x+2)}$
Ex 7.5 Q2 $\frac{1}{x^2-9}$
Ex 7.5 Q3 $\frac{3x-1}{(x-1)(x-2)(x-3)}$
Ex 7.5 Q4 $\frac{x}{(x-1)(x-2)(x-3)}$
Ex 7.5 Q5 $\frac{2x}{x^2+3x+2}$
Ex 7.5 Q6 $\frac{1-x^2}{x(1-2x)}$
Ex 7.5 Q7 $\frac{x}{(x^2+1)(x-1)}$
Ex 7.5 Q8 $\frac{x}{(x-1)^2(x+2)}$
Ex 7.5 Q9 $\frac{3x+5}{x^3-x^2-x+1}$
Ex 7.5 Q10 $\frac{2x-3}{(x^2-1)(2x+3)}$
Ex 7.5 Q11 $\frac{5x}{(x+1)(x^2-4)}$
Ex 7.5 Q12 $\frac{x^3+x+1}{x^2-1}$
Ex 7.5 Q13 $\frac{2}{(1-x)(1+x^2)}$
Ex 7.5 Q14 $\frac{3x-1}{(x+2)^2}$
Ex 7.5 Q15 $\frac{1}{x^4-1}$
Ex 7.5 Q16 $\frac{1}{x(x^n+1)}$
Ex 7.5 Q17 $\frac{\cos x}{(1-\sin x)(2-\sin x)}$
Ex 7.5 Q18 $\frac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}$
Ex 7.5 Q19 $\frac{2x}{(x^2+1)(x^2+3)}$
Ex 7.5 Q20 $\frac{1}{x(x^4-1)}$
Ex 7.5 Q21 $\frac{1}{e^x-1}$
Ex 7.5 Q22 $\int \frac{x\,dx}{(x-1)(x-2)}$
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Ex 7.5 Q23 $\int \frac{dx}{x(x^2+1)}$
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7.6 Integration by Parts

31 q

Solved Examples

Worked · 7

7.6 Eg.17 $\int x \cos x \, dx$
7.6 Eg.18 $\int \log x \, dx$
7.6 Eg.19 $\int x\, e^{x} \, dx$
7.6 Eg.20 $\int \frac{x \sin^{-1} x}{\sqrt{1-x^{2}}} \, dx$
7.6 Eg.21 $\int e^{x} \sin x \, dx$
Find the following integrals.
7.6 Eg.22(i) $\int e^{x}\left(\tan^{-1} x + \frac{1}{1+x^{2}}\right) dx$
7.6 Eg.22(ii) $\int \frac{(x^{2}+1)\, e^{x}}{(x+1)^{2}} \, dx$

Exercise 7.6

Practice · 24

Ex 7.6 Q1 $x \sin x$
Ex 7.6 Q2 $x \sin 3x$
Ex 7.6 Q3 $x^{2} e^{x}$
Ex 7.6 Q4 $x \log x$
Ex 7.6 Q5 $x \log 2x$
Ex 7.6 Q6 $x^{2} \log x$
Ex 7.6 Q7 $x \sin^{-1} x$
Ex 7.6 Q8 $x \tan^{-1} x$
Ex 7.6 Q9 $x \cos^{-1} x$
Ex 7.6 Q10 $(\sin^{-1} x)^{2}$
Ex 7.6 Q11 $\dfrac{x \cos^{-1} x}{\sqrt{1-x^{2}}}$
Ex 7.6 Q12 $x \sec^{2} x$
Ex 7.6 Q13 $\tan^{-1} x$
Ex 7.6 Q14 $x (\log x)^{2}$
Ex 7.6 Q15 $(x^{2}+1) \log x$
Ex 7.6 Q16 $e^{x} (\sin x + \cos x)$
Ex 7.6 Q17 $\dfrac{x\, e^{x}}{(1+x)^{2}}$
Ex 7.6 Q18 $e^{x}\left(\dfrac{1+\sin x}{1+\cos x}\right)$
Ex 7.6 Q19 $e^{x}\left(\dfrac{1}{x} - \dfrac{1}{x^{2}}\right)$
Ex 7.6 Q20 $\dfrac{(x-3)\, e^{x}}{(x-1)^{3}}$
Ex 7.6 Q21 $e^{2x} \sin x$
Ex 7.6 Q22 $\sin^{-1}\left(\dfrac{2x}{1+x^{2}}\right)$
Ex 7.6 Q23 $\int x^{2} e^{x^{3}} \, dx$
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Ex 7.6 Q24 $\int e^{x} \sec x\, (1 + \tan x) \, dx$
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7.6 Integrals of Special Forms

13 q

Solved Examples

Worked · 2

7.7 Eg.23 $\int \sqrt{x^2 + 2x + 5}\,dx$
7.7 Eg.24 $\int \sqrt{3 - 2x - x^2}\,dx$

Exercise 7.7

Practice · 11

Ex 7.7 Q1 $\sqrt{4 - x^2}$
Ex 7.7 Q2 $\sqrt{1 - 4x^2}$
Ex 7.7 Q3 $\sqrt{x^2 + 4x + 6}$
Ex 7.7 Q4 $\sqrt{x^2 + 4x + 1}$
Ex 7.7 Q5 $\sqrt{1 - 4x - x^2}$
Ex 7.7 Q6 $\sqrt{x^2 + 4x - 5}$
Ex 7.7 Q7 $\sqrt{1 + 3x - x^2}$
Ex 7.7 Q8 $\sqrt{x^2 + 3x}$
Ex 7.7 Q9 $\sqrt{1 + \dfrac{x^2}{9}}$
Ex 7.7 Q10 $\int \sqrt{1 + x^2}\,dx$
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Ex 7.7 Q11 $\int \sqrt{x^2 - 8x + 7}\,dx$
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7.7 Definite Integral and the Fundamental Theorem

26 q

Solved Examples

Worked · 4

Evaluate the following integrals:
7.8 Eg.25(i) $\int_{2}^{3} x^{2}\,dx$
7.8 Eg.25(ii) $\int_{4}^{9} \frac{\sqrt{x}}{\left(30 - x^{\frac{3}{2}}\right)^{2}}\,dx$
7.8 Eg.25(iii) $\int_{1}^{2} \frac{x\,dx}{(x+1)(x+2)}$
7.8 Eg.25(iv) $\int_{0}^{\frac{\pi}{4}} \sin^{3} 2t \cos 2t \,dt$

Exercise 7.8

Practice · 22

Ex 7.8 Q1 $\int_{-1}^{1} (x+1)\,dx$
Ex 7.8 Q2 $\int_{2}^{3} \frac{1}{x}\,dx$
Ex 7.8 Q3 $\int_{1}^{2} (4x^{3} - 5x^{2} + 6x + 9)\,dx$
Ex 7.8 Q4 $\int_{0}^{\frac{\pi}{4}} \sin 2x \,dx$
Ex 7.8 Q5 $\int_{0}^{\frac{\pi}{2}} \cos 2x \,dx$
Ex 7.8 Q6 $\int_{4}^{5} e^{x}\,dx$
Ex 7.8 Q7 $\int_{0}^{\frac{\pi}{4}} \tan x \,dx$
Ex 7.8 Q8 $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \operatorname{cosec} x \,dx$
Ex 7.8 Q9 $\int_{0}^{1} \frac{dx}{\sqrt{1 - x^{2}}}$
Ex 7.8 Q10 $\int_{0}^{1} \frac{dx}{1 + x^{2}}$
Ex 7.8 Q11 $\int_{2}^{3} \frac{dx}{x^{2} - 1}$
Ex 7.8 Q12 $\int_{0}^{\frac{\pi}{2}} \cos^{2} x \,dx$
Ex 7.8 Q13 $\int_{2}^{3} \frac{x\,dx}{x^{2} + 1}$
Ex 7.8 Q14 $\int_{0}^{1} \frac{2x + 3}{5x^{2} + 1}\,dx$
Ex 7.8 Q15 $\int_{0}^{1} x\, e^{x^{2}}\,dx$
Ex 7.8 Q16 $\int_{1}^{2} \frac{5x^{2}}{x^{2} + 4x + 3}\,dx$
Ex 7.8 Q17 $\int_{0}^{\frac{\pi}{4}} (2\sec^{2} x + x^{3} + 2)\,dx$
Ex 7.8 Q18 $\int_{0}^{\pi} \left(\sin^{2}\frac{x}{2} - \cos^{2}\frac{x}{2}\right)\,dx$
Ex 7.8 Q19 $\int_{0}^{2} \frac{6x + 3}{x^{2} + 4}\,dx$
Ex 7.8 Q20 $\int_{0}^{1} \left(x\, e^{x} + \sin\frac{\pi x}{4}\right)\,dx$
Ex 7.8 Q21 $\int_{1}^{\sqrt{3}} \frac{dx}{1 + x^{2}}$
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Ex 7.8 Q22 $\int_{0}^{\frac{2}{3}} \frac{dx}{4 + 9x^{2}}$
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7.9 Evaluation of Definite Integrals by Substitution

12 q

Solved Examples

Worked · 2

7.9 Eg.26 $\int_{-1}^{1} 5x^{4}\sqrt{x^{5}+1}\,dx$
7.9 Eg.27 $\int_{0}^{1} \frac{\tan^{-1} x}{1+x^{2}}\,dx$

Exercise 7.9

Practice · 10

Ex 7.9 Q1 $\int_{0}^{1} \frac{x}{x^{2}+1}\,dx$
Ex 7.9 Q2 $\int_{0}^{\pi/2} \sqrt{\sin\phi}\,\cos^{5}\phi\,d\phi$
Ex 7.9 Q3 $\int_{0}^{1} \sin^{-1}\left(\frac{2x}{1+x^{2}}\right)dx$
Ex 7.9 Q4 $\int_{0}^{2} x\sqrt{x+2}\,dx$
Ex 7.9 Q5 $\int_{0}^{\pi/2} \frac{\sin x}{1+\cos^{2} x}\,dx$
Ex 7.9 Q6 $\int_{0}^{2} \frac{dx}{x+4-x^{2}}$
Ex 7.9 Q7 $\int_{-1}^{1} \frac{dx}{x^{2}+2x+5}$
Ex 7.9 Q8 $\int_{1}^{2} \left(\frac{1}{x} - \frac{1}{2x^{2}}\right) e^{2x}\,dx$
Ex 7.9 Q9 $\int_{1/3}^{1} \frac{(x-x^{3})^{1/3}}{x^{4}}\,dx$
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Ex 7.9 Q10 $f(x) = \int_{0}^{x} t\sin t\,dt$
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7.10 Some Properties of Definite Integrals

28 q

Solved Examples

Worked · 7

7.10 Eg.28 $\int_{-1}^{2} |x^3 - x|\,dx$
7.10 Eg.29 $\int_{-\pi/4}^{\pi/4} \sin^2 x\,dx$
7.10 Eg.30 $\int_{0}^{\pi} \frac{x \sin x}{1 + \cos^2 x}\,dx$
7.10 Eg.31 $\int_{-1}^{1} \sin^5 x \cos^4 x\,dx$
7.10 Eg.32 $\int_{0}^{\pi/2} \frac{\sin^4 x}{\sin^4 x + \cos^4 x}\,dx$
7.10 Eg.33 $\int_{\pi/6}^{\pi/3} \frac{dx}{1 + \sqrt{\tan x}}$
7.10 Eg.34 $\int_{0}^{\pi/2} \log \sin x\,dx$

Exercise 7.10

Practice · 21

Ex 7.10 Q1 $\int_{0}^{\pi/2} \cos^2 x\,dx$
Ex 7.10 Q2 $\int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}\,dx$
Ex 7.10 Q3 $\int_{0}^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x}\,dx$
Ex 7.10 Q4 $\int_{0}^{\pi/2} \frac{\cos^5 x}{\sin^5 x + \cos^5 x}\,dx$
Ex 7.10 Q5 $\int_{-5}^{5} |x + 2|\,dx$
Ex 7.10 Q6 $\int_{2}^{8} |x - 5|\,dx$
Ex 7.10 Q7 $\int_{0}^{1} x(1 - x)^n\,dx$
Ex 7.10 Q8 $\int_{0}^{\pi/4} \log(1 + \tan x)\,dx$
Ex 7.10 Q9 $\int_{0}^{2} x\sqrt{2 - x}\,dx$
Ex 7.10 Q10 $\int_{0}^{\pi/2}(2\log \sin x - \log \sin 2x)\,dx$
Ex 7.10 Q11 $\int_{-\pi/2}^{\pi/2} \sin^2 x\,dx$
Ex 7.10 Q12 $\int_{0}^{\pi} \frac{x\,dx}{1 + \sin x}$
Ex 7.10 Q13 $\int_{-\pi/2}^{\pi/2} \sin^7 x\,dx$
Ex 7.10 Q14 $\int_{0}^{2\pi} \cos^5 x\,dx$
Ex 7.10 Q15 $\int_{0}^{\pi/2} \frac{\sin x - \cos x}{1 + \sin x \cos x}\,dx$
Ex 7.10 Q16 $\int_{0}^{\pi} \log(1 + \cos x)\,dx$
Ex 7.10 Q17 $\int_{0}^{a} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}}\,dx$
Ex 7.10 Q18 $\int_{0}^{4} |x - 1|\,dx$
Ex 7.10 Q19 $\int_{0}^{a} f(x)g(x)\,dx = 2\int_{0}^{a} f(x)\,dx$
Ex 7.10 Q20 $\int_{-\pi/2}^{\pi/2}(x^3 + x \cos x + \tan^5 x + 1)\,dx$
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Ex 7.10 Q21 $\int_{0}^{\pi/2} \log\left(\frac{4 + 3\sin x}{4 + 3\cos x}\right)\,dx$
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Miscellaneous Exercise on Chapter 7

48 q

Solved Examples

Worked · 8

Misc Eg.35 $\int \cos 6x\, \sqrt{1+\sin 6x}\; dx$
Misc Eg.36 $\int \frac{(x^4 - x)^{\frac{1}{4}}}{x^5}\, dx$
Misc Eg.37 $\int \frac{x^4\, dx}{(x-1)(x^2+1)}$
Misc Eg.38 $\int \left[\log(\log x) + \frac{1}{(\log x)^2}\right] dx$
Misc Eg.39 $\int \left[\sqrt{\cot x} + \sqrt{\tan x}\right] dx$
Misc Eg.40 $\int \frac{\sin 2x \cos 2x\, dx}{\sqrt{9 - \cos^4(2x)}}$
Misc Eg.41 $\int_{-1}^{\frac{3}{2}} |x \sin(\pi x)|\, dx$
Misc Eg.42 $\int_{0}^{\pi} \frac{x\, dx}{a^2\cos^2 x + b^2\sin^2 x}$

Miscellaneous Exercise

Practice · 40

Misc Q1 $\frac{1}{x - x^3}$
Misc Q2 $\frac{1}{\sqrt{x+a} + \sqrt{x+b}}$
Misc Q3 $\frac{1}{x\sqrt{ax - x^2}}$
Misc Q4 $\frac{1}{x^2 (x^4 + 1)^{\frac{3}{4}}}$
Misc Q5 $\frac{1}{x^{\frac{1}{2}} + x^{\frac{1}{3}}}$
Misc Q6 $\frac{5x}{(x+1)(x^2+9)}$
Misc Q7 $\frac{\sin x}{\sin(x - a)}$
Misc Q8 $\frac{e^{5\log x} - e^{4\log x}}{e^{3\log x} - e^{2\log x}}$
Misc Q9 $\frac{\cos x}{\sqrt{4 - \sin^2 x}}$
Misc Q10 $\frac{\sin^8 x - \cos^8 x}{1 - 2\sin^2 x \cos^2 x}$
Misc Q11 $\frac{1}{\cos(x + a)\cos(x + b)}$
Misc Q12 $\frac{x^3}{\sqrt{1 - x^8}}$
Misc Q13 $\frac{e^x}{(1 + e^x)(2 + e^x)}$
Misc Q14 $\frac{1}{(x^2 + 1)(x^2 + 4)}$
Misc Q15 $\cos^3 x\; e^{\log \sin x}$
Misc Q16 $e^{3\log x}(x^4 + 1)^{-1}$
Misc Q17 $f'(ax + b)\,[f(ax + b)]^n$
Misc Q18 $\frac{1}{\sqrt{\sin^3 x \, \sin(x + \alpha)}}$
Misc Q19 $\sqrt{\frac{1 - \sqrt{x}}{1 + \sqrt{x}}}$
Misc Q20 $\frac{2 + \sin 2x}{1 + \cos 2x}\, e^x$
Misc Q21 $\frac{x^2 + x + 1}{(x + 1)^2 (x + 2)}$
Misc Q22 $\tan^{-1}\sqrt{\frac{1 - x}{1 + x}}$
Misc Q23 $\frac{\sqrt{x^2 + 1}\,\left[\log(x^2 + 1) - 2\log x\right]}{x^4}$
Misc Q24 $\int_{\frac{\pi}{2}}^{\pi} e^x\left(\frac{1 - \sin x}{1 - \cos x}\right) dx$
Misc Q25 $\int_{0}^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos^4 x + \sin^4 x}\, dx$
Misc Q26 $\int_{0}^{\frac{\pi}{2}} \frac{\cos^2 x\, dx}{\cos^2 x + 4\sin^2 x}$
Misc Q27 $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x + \cos x}{\sqrt{\sin 2x}}\, dx$
Misc Q28 $\int_{0}^{1} \frac{dx}{\sqrt{1 + x} - \sqrt{x}}$
Misc Q29 $\int_{0}^{\frac{\pi}{4}} \frac{\sin x + \cos x}{9 + 16\sin 2x}\, dx$
Misc Q30 $\int_{0}^{\frac{\pi}{2}} \sin 2x \, \tan^{-1}(\sin x)\, dx$
Misc Q31 $\int_{1}^{4} \left[|x - 1| + |x - 2| + |x - 3|\right] dx$
Misc Q32 $\int_{1}^{3} \frac{dx}{x^2 (x + 1)} = \frac{2}{3} + \log \frac{2}{3}$
Misc Q33 $\int_{0}^{1} x\, e^x\, dx = 1$
Misc Q34 $\int_{-1}^{1} x^{17} \cos^4 x\, dx = 0$
Misc Q35 $\int_{0}^{\frac{\pi}{2}} \sin^3 x\, dx = \frac{2}{3}$
Misc Q36 $\int_{0}^{\frac{\pi}{4}} 2\tan^3 x\, dx = 1 - \log 2$
Misc Q37 $\int_{0}^{1} \sin^{-1} x\, dx = \frac{\pi}{2} - 1$
Misc Q38 $\int \frac{dx}{e^x + e^{-x}}$
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Misc Q39 $\int \frac{\cos 2x}{(\sin x + \cos x)^2}\, dx$
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Misc Q40 $f(a + b - x) = f(x)$
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What the board has asked from this chapter

Questions per paper · 26 papers · 2022 to 2026

Bars average the full papers of each year. 108 further questions come from set variants and are listed below.

2026

40 q

1 $\displaystyle\int \dfrac{dx}{1 + \cos x}$
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Integration using Trigonometric Identities
3 $\int \dfrac{dx}{\sqrt{e^{-2x} - 1}}$
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Integration by Substitution
8 $\int \frac{3ax}{b^2 + c^2x^2}\,dx = A \log \left| b^2 + c^2x^2 \right| + K$
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Integration by Substitution
9 $\int_{-1}^{1} \frac{x^3}{x^2 + 2\left| x \right| + 1}\,dx$
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Properties of Definite Integrals
9 $\displaystyle\int \dfrac{dx}{\sqrt{25 - 16x^2}}$
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Integrals of Some Particular Functions
9 $\displaystyle\int_{0}^{2a} \dfrac{1}{1 + 4x^2}\, dx = \dfrac{\pi}{6}$
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Definite Integrals and the Fundamental Theorem
10 $\displaystyle\int \dfrac{1}{\sqrt{1 + \cos 2x}}\,dx$
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Integration using Trigonometric Identities
10 $\displaystyle\int \sqrt{\dfrac{1 - \cos 2x}{1 + \cos 2x}}\,dx$
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Integration using Trigonometric Identities
10 $\displaystyle\int_0^1 \dfrac{dx}{e^x + e^{-x}} = \tan^{-1} e + k$
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Definite Integrals by Substitution
10 $\displaystyle\int \sqrt{\dfrac{1 + \cos x}{1 - \cos x}}\,dx$
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Integration using Trigonometric Identities
10 $\int \dfrac{dx}{2^x + 2^{-x}}$
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Integration by Substitution
11 $\displaystyle\int_{0}^{1} (6x^2 - 4x + k)\,dx = 0$
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Definite Integrals and the Fundamental Theorem
11 $\displaystyle\int_{-5}^{-1} \dfrac{1}{x}\,dx$
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Definite Integrals and the Fundamental Theorem
11 $\displaystyle\int_{0}^{1} \dfrac{1}{3x - 4}\,dx$
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Definite Integrals and the Fundamental Theorem
11 $\int_{-1}^{1} (1 - |x|)\, dx$
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Properties of Definite Integrals
12 $\displaystyle\int \dfrac{dx}{\sec x + \tan x}$
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Integration using Trigonometric Identities
14 $\int \dfrac{\cos x}{\sqrt{\sin^2 x + 1}}\, dx$
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Integrals of Some Particular Functions
26 (a) $\int \dfrac{dx}{x^{1/2} + x^{1/3}}$
Integration by Substitution
26 $\int_{0}^{1} x \tan^{-1} x \, dx$
Integration by Parts
26 $\displaystyle\int \dfrac{x + 2}{\sqrt{9x - x^2}}\, dx$
Integrals of Some Particular Functions
27 (a) $\displaystyle\int \dfrac{x - \sin x}{1 - \cos x}\,dx$
Integration by Parts
27 (a) $\displaystyle\int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \dfrac{dx}{1 + \sqrt{\cot x}}$
Properties of Definite Integrals
26 (b) $\int \tan^{-1} \left(\dfrac{1 - x}{1 + x}\right) dx$
Integration by Parts
27 (a) $\int \sqrt{\frac{x + 2}{x - 2}} \, dx$
Integrals of Some Particular Functions
26 $\int_{0}^{1} \log(1 + x^2)\,dx$
Integration by Parts
27 (b) $\int \frac{x^2}{(x^2 + 9) \, (x^2 + 16)} \, dx$
Integration by Partial Fractions
27 (b) $\displaystyle\int_{0}^{2} \dfrac{1}{\sqrt{x^2 + 2x + 3}}\,dx$
Integrals of Some Particular Functions
27 $\int_{0}^{\pi} \dfrac{\sin^{2026} x}{\sin^{2026} x + \cos^{2026} x}\, dx$
Properties of Definite Integrals
27 (b) $\displaystyle\int_{\frac{-\pi}{6}}^{\frac{\pi}{2}} (\sin |x| + \cos |x|)\, dx$
Definite Integrals and the Fundamental Theorem
27 $\displaystyle\int \dfrac{3x - 1}{\sqrt{x^2 - 4x}}\, dx$
Integrals of Some Particular Functions
28 $\dfrac{d}{dx}(F(x)) = \dfrac{1}{e^x + 1}$
Integration by Substitution
28 $\int_{0}^{\pi/2} \dfrac{\sin^2 x}{1 + \sin 2x}\, dx$
Properties of Definite Integrals
28 $I_1 = \int_{-\pi/4}^{\pi/4} \frac{dx}{1 + \cos 2x}$
Properties of Definite Integrals
28 (a) $\int \dfrac{\cos x}{(2 + \sin x)\,(4 + \sin x)}\, dx$
Integration by Partial Fractions
28 (b) $\int \dfrac{x + 3}{x^2 + 4x + 5}\, dx$
Integrals of Some Particular Functions
30 $\displaystyle\int \dfrac{2x + 1}{\sqrt{6x + x^2}}\, dx$
Integrals of Some Particular Functions
30 $\int_{0}^{1} x \sin^{-1} x \, dx$
Integration by Parts
31 $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \dfrac{\cos^2 x}{2^x + 1}\, dx$
Properties of Definite Integrals
32 (b) $\displaystyle\int_0^1 \dfrac{x \tan^{-1} x}{\left(1 + x^2\right)^{3/2}}\, dx$
Integration by Parts
A racing track is build around an elliptical ground whose equation is given by $9x^2 + 16y^2 = 144$ . The width of the track is 3 m as shown below : Based on given information, answer the following questions :
37 (ii) $x$
Integrals of Special Forms

2025

59 q

3 $\displaystyle\int_0^{a} x\,dx \le \dfrac{a}{2} + 6$
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Definite Integrals and the Fundamental Theorem
11 $\displaystyle\int \dfrac{\cos 2x}{\sin^2 x \cos^2 x}\ dx$
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Integration using Trigonometric Identities
3 $\displaystyle\int \dfrac{dx}{\sin^2 x \cos^2 x}$
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Integration using Trigonometric Identities
3 $f(2a - x) = f(x)$
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Properties of Definite Integrals
3 $\int_{-1}^{1} \frac{|x|}{x}\,dx$
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Properties of Definite Integrals
3 $\int_{0}^{1} \frac{e^x}{1 + x}\,dx = \alpha$
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Integration by Parts
3 $\int \frac{1 - 2\sin x}{\cos^2 x}\,dx$
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Integration using Trigonometric Identities
12 $\displaystyle\int_{0}^{a} \dfrac{1}{1 + 4x^2}\ dx = \dfrac{\pi}{8}$
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Definite Integrals and the Fundamental Theorem
11 $\displaystyle\int \dfrac{\cos 2x - \cos 2\theta}{\cos x - \cos \theta}\ dx$
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Integration using Trigonometric Identities
12 $\displaystyle\int_{0}^{1} \dfrac{2x}{5x^2 + 1}\ dx$
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Definite Integrals by Substitution
10 $f'(x) = 3(x^2 + 2x) - \dfrac{4}{x^3} + 5$
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Integration as the Inverse of Differentiation
10 $\int \frac{2^{\frac{1}{x}}}{x^2}\,dx = k \cdot 2^{\frac{1}{x}} + C$
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Integration by Substitution
11 $\displaystyle\int \dfrac{x + 5}{(x + 6)^2}\, e^x\, dx$
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Integrals of Special Forms
11 $\displaystyle\int \sqrt{1 + \sin x}\ dx$
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Integration using Trigonometric Identities
12 $\displaystyle\int \dfrac{e^{9\log x} - e^{8\log x}}{e^{6\log x} - e^{5\log x}}\, dx$
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Integration as the Inverse of Differentiation
12 $\displaystyle\int_{0}^{\pi/2} \cos x \cdot e^{\sin x}\ dx$
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Definite Integrals by Substitution
13 $For a function f(x), which of the following holds true ?$
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Properties of Definite Integrals
14 $\int e^x (\cos x - \sin x)\, dx$
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Integration by Parts
14 $\displaystyle\int \dfrac{a^x}{\sqrt{1 - a^{2x}}}\, dx$
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Integrals of Some Particular Functions
14 $\int e^{-3\log x}\, dx = f(x) + C$
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Integration as the Inverse of Differentiation
14 $\displaystyle\int \dfrac{e^{-x}}{16 + 9e^{-2x}}\, dx$
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Integrals of Some Particular Functions
14 $\displaystyle\int \dfrac{e^x}{\sqrt{4 - e^{2x}}}\, dx$
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Integrals of Some Particular Functions
14 $\int \frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha}\, dx$
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Integration using Trigonometric Identities
15 $\int\limits_0^1 \frac{dx}{e^x + e^{-x}}$
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Definite Integrals by Substitution
22 $\displaystyle\int 2x^3\,e^{x^2}\,dx$
Integration by Parts
22 $\displaystyle\int_0^{\pi} \dfrac{\sin 2px}{\sin x}\;dx$
Properties of Definite Integrals
22 $\displaystyle\int_0^{\pi/4} \sqrt{1 + \sin 2x}\;dx$
Integration using Trigonometric Identities
26 $\int \frac{\sqrt{x}}{1 + \sqrt{x^{3/2}}}\, dx$
Integration by Substitution
27 (a) $\int \frac{2x - 1}{(x - 1)(x + 2)(x - 3)}\,dx$
Integration by Partial Fractions
27 (a) $\displaystyle\int \dfrac{2x}{(x^2 + 3)(x^2 - 5)}\ dx$
Integration by Partial Fractions
27 (b) $\int_{0}^{5} (|x - 1| + |x - 2| + |x - 5|)\,dx$
Definite Integrals and the Fundamental Theorem
27 (b) $\displaystyle\int_{1}^{4} \left(|x - 2| + |x - 4|\right)\ dx$
Properties of Definite Integrals
28 (a) $\int \frac{x + \sin x}{1 + \cos x}\,dx$
Integration using Trigonometric Identities
28 (b) $\int_{0}^{\frac{\pi}{4}} \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}}$
Definite Integrals by Substitution
28 $\int \frac{\cos x\, dx}{1 + \cos x + \sin x}$
Integration using Trigonometric Identities
28 $\int\limits_{\pi/2}^{\pi} e^x \left(\frac{1 - \sin x}{1 - \cos x}\right) dx$
Integration by Parts
30 (a) $\int \frac{\cos 2x}{(\sin x + \cos x)^2}\,dx$
Integration using Trigonometric Identities
30 (b) $\int_{0}^{\frac{\pi}{2}} \frac{5\sin x + 3\cos x}{\sin x + \cos x}\,dx$
Properties of Definite Integrals
31 $\displaystyle\int_a^{b} x^3\,dx = 0$
Definite Integrals and the Fundamental Theorem
31 $f(a - x) = f(x)$
Properties of Definite Integrals
31 $\displaystyle\int \dfrac{1}{x} \sqrt{\dfrac{x + a}{x - a}}\;dx$
Integrals of Some Particular Functions
32 (a) $\displaystyle\int \dfrac{x^2 + 1}{(x - 1)^2 (x + 3)}\, dx$
Integration by Partial Fractions
32 (a) $\displaystyle\int \dfrac{x}{(x - 1)(x^2 + 4)}\, dx$
Integration by Partial Fractions
32 (a) $\displaystyle\int \dfrac{3x + 1}{(x - 2)^2 (x + 2)}\, dx$
Integration by Partial Fractions
32 (a) $\displaystyle\int \dfrac{x^2 + 1}{(x^2 + 2)(2x^2 + 1)}\, dx$
Integration by Partial Fractions
32 (b) $\displaystyle\int_{0}^{\pi/2} \dfrac{x}{\cos x + \sin x}\, dx$
Properties of Definite Integrals
32 (b) $\displaystyle\int_{0}^{\pi} \dfrac{x \sin x}{1 + \cos^2 x}\, dx$
Properties of Definite Integrals
33 $\displaystyle\int \dfrac{x^2 + x + 1}{(x + 2)(x^2 + 1)}\;dx$
Integration by Partial Fractions
33 $\int \sin^{-1} \sqrt{\frac{x}{a + x}}\,dx$
Integration by Parts
32 (b) $\displaystyle\int_{0}^{\pi/2} \dfrac{x}{\sin x + \cos x}\, dx$
Properties of Definite Integrals
32 (b) $\displaystyle\int_{0}^{\pi} \dfrac{x \tan x}{\sec x + \tan x}\, dx$
Properties of Definite Integrals
33 $\displaystyle\int \left(\sqrt{\tan x} + \sqrt{\cot x}\right) dx$
Integrals of Special Forms
34 (a) $\int \frac{\cos x}{(4 + \sin^2 x)(5 - 4\cos^2 x)}\, dx$
Integration by Partial Fractions
34 (a) $\int\limits_0^{\pi/4} \frac{\sin x \cos x}{\cos^4 x + \sin^4 x}\, dx$
Definite Integrals by Substitution
33 $\displaystyle\int \dfrac{5x}{(x + 1)(x^2 + 9)}\;dx$
Integration by Partial Fractions
34 (b) $\int \frac{\sqrt{x^2 + 1}\left[\log(x^2 + 1) - 2\log x\right]}{x^2}\, dx$
Integration by Parts
34 (a) $\int\limits_0^{3/2} |x \cos \pi x|\, dx$
Definite Integrals and the Fundamental Theorem
34 (b) $\int\limits_0^{\pi} \frac{dx}{a^2 \cos^2 x + b^2 \sin^2 x}$
Properties of Definite Integrals
34 (b) $\int \frac{dx}{\sin x + \sin 2x}$
Integration by Partial Fractions

2024

52 q

6 $\int_a^b f(x)\,dx$
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Properties of Definite Integrals
6 $\displaystyle\int_0^{\pi/2} \dfrac{\sin x - \cos x}{1 + \sin x \cos x}\, dx$
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Properties of Definite Integrals
6 $\int\limits_{-a}^{a} f(x)\, dx = 0$
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Properties of Definite Integrals
8 $\displaystyle\int_{-1}^{1} |x|\ dx$
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Definite Integrals and the Fundamental Theorem
8 $\displaystyle\int_{0}^{\pi} \tan^{2}\left(\dfrac{\theta}{3}\right) d\theta$
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Definite Integrals and the Fundamental Theorem
8 $\displaystyle\int_{0}^{3} \dfrac{dx}{\sqrt{9 - x^2}}$
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Definite Integrals and the Fundamental Theorem
11 $\displaystyle\int \dfrac{1}{x(\log x)^2}\, dx$
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Integration by Substitution
11 $\displaystyle\int_{\pi/4}^{\pi/2} \cot\theta \; \mathrm{cosec}^2\theta \; d\theta$
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Definite Integrals by Substitution
11 $\displaystyle\int_{-2}^{3} x^2\, \mathrm{d}x = \mathrm{k}\int_{0}^{2} x^2\, \mathrm{d}x + \int_{2}^{3} x^2\, \mathrm{d}x$
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Properties of Definite Integrals
12 $\sqrt{1 + \sin 2x}$
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Integration using Trigonometric Identities
12 $\displaystyle\int_{1}^{e} \log x\, \mathrm{d}x$
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Integration by Parts
12 $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(x) \cos^3 x \; dx$
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Properties of Definite Integrals
12 $\displaystyle\int \dfrac{dx}{\sqrt{9 - 4x^2}}$
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Integrals of Some Particular Functions
12 $\displaystyle\int_{-1}^{1} x\,|x|\, dx$
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Properties of Definite Integrals
12 $g(x)$
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Properties of Definite Integrals
18 $\displaystyle\int_0^2 2 e^{2x}\, dx = \int_0^a e^x\, dx$
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Definite Integrals and the Fundamental Theorem
18 $\int_0^a \frac{1}{4 + x^2} \; dx = \frac{\pi}{6}$
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Definite Integrals and the Fundamental Theorem
22 $\displaystyle\int_0^{a^3} \dfrac{x^2}{x^6 + a^6}\, dx$
Definite Integrals by Substitution
23 (a) $\int\limits_{0}^{\pi/2} \sin 2x \cos 3x\, dx$
Integration using Trigonometric Identities
23 (a) $\int x \sqrt{1 + 2x} \; dx$
Integration by Substitution
24 (a) $\displaystyle\int \cos^3 x\; \mathrm{e}^{\log \sin x}\, \mathrm{d}x$
Integration by Substitution
23 (b) $\dfrac{d}{dx} F(x) = \dfrac{1}{\sqrt{2x - x^2}}$
Integrals of Some Particular Functions
23 (b) $\int_0^{\frac{\pi^2}{4}} \frac{\sin \sqrt{x}}{\sqrt{x}} \; dx$
Definite Integrals by Substitution
24 (b) $\displaystyle\int \dfrac{1}{5 + 4x - x^2}\, \mathrm{d}x$
Integrals of Some Particular Functions
24 $\displaystyle\int \dfrac{e^{4x} - 1}{e^{4x} + 1}\, dx$
Integration by Substitution
25 $\displaystyle\int_{-1/2}^{1/2} \cos x \cdot \log\left(\dfrac{1+x}{1-x}\right) dx$
Properties of Definite Integrals
25 $\displaystyle\int \dfrac{2x}{(x^{2} + 1)(x^{2} - 4)}\,dx$
Integration by Partial Fractions
25 $\displaystyle\int \dfrac{1}{x(x^{2} - 1)}\,dx$
Integration by Substitution
26 $\int \dfrac{x^{-1}}{(\log x)^2 - 5 \log x + 4}\, dx$
Integration by Substitution
27 $\displaystyle\int x^2 \log (x^2 - 1)\, dx$
Integration by Parts
27 (a) $\displaystyle\int_{0}^{\pi/4} \dfrac{x\,dx}{1 + \cos 2x + \sin 2x}$
Properties of Definite Integrals
27 (b) $\displaystyle\int e^{x}\left[\dfrac{1}{(1 + x^{2})^{3/2}} + \dfrac{x}{\sqrt{1 + x^{2}}}\right] dx$
Integrals of Special Forms
27 (a) $\displaystyle\int_{-2}^{2} \sqrt{\dfrac{2 - x}{2 + x}}\, dx$
Definite Integrals by Substitution
28 $\displaystyle\int \dfrac{2x + 3}{x^{2}(x + 3)}\,dx$
Integration by Partial Fractions
28 $\displaystyle\int \sec^{3}\theta\,d\theta$
Integration by Parts
27 (b) $\displaystyle\int \dfrac{1}{x\,[(\log x)^2 - 3 \log x - 4]}\, dx$
Integration by Partial Fractions
28 $\displaystyle\int \dfrac{3x + 5}{\sqrt{x^{2} + 2x + 4}}\,dx$
Integrals of Some Particular Functions
28 (a) $\displaystyle\int_{0}^{\pi} \dfrac{\mathrm{e}^{\cos x}}{\mathrm{e}^{\cos x} + \mathrm{e}^{-\cos x}}\, \mathrm{d}x$
Properties of Definite Integrals
28 $\int \dfrac{x^2 + 1}{(x^2 + 2)(x^2 + 4)}\, dx$
Integration by Partial Fractions
28 (b) $\displaystyle\int \dfrac{2x + 1}{(x + 1)^2\,(x - 1)}\, \mathrm{d}x$
Integration by Partial Fractions
28 (a) $\int \frac{x^2}{(x^2 + 4)\,(x^2 + 9)} \; dx$
Integration by Partial Fractions
29 (a) $\int \dfrac{2 + \sin 2x}{1 + \cos 2x}\, e^x\, dx$
Integration by Parts
28 (b) $\int_1^3 \left( |x - 1| + |x - 2| + |x - 3| \right) dx$
Definite Integrals and the Fundamental Theorem
29 (b) $\int\limits_{0}^{\pi/4} \dfrac{1}{\sin x + \cos x}\, dx$
Integration using Trigonometric Identities
30 $\int \dfrac{\sqrt{x}}{(x + 1)(x - 1)}\, dx$
Integration by Partial Fractions
31 $\displaystyle\int x^2 \cdot \sin^{-1}(x^{3/2})\, dx$
Integration by Parts
32 (a) $\int_0^{\frac{\pi}{2}} e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx$
Integration by Parts
32 (b) $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x + \cos x}{\sqrt{\sin 2x}} \; dx$
Definite Integrals by Substitution
33 (a) $\int_0^{\frac{\pi}{4}} \frac{\sin x + \cos x}{9 + 16 \sin 2x} \; dx$
Definite Integrals by Substitution
33 (b) $\int_0^{\frac{\pi}{2}} \sin 2x \, \tan^{-1} (\sin x) \; dx$
Integration by Parts
34 (a) $\int \frac{(3 \cos x - 2) \sin x}{5 - \sin^2 x - 4 \cos x} \; dx$
Integration by Substitution
34 (b) $\int_{-2}^{2} \frac{x^3 + |x| + 1}{x^2 + 4|x| + 4} \; dx$
Properties of Definite Integrals

2023

58 q

1 $\frac{d}{dx} f(x) = 2x + \frac{3}{x}$
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Integration as the Inverse of Differentiation
1 $\int \frac{1 + \tan x}{1 - \tan x} \, dx$
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Integration using Trigonometric Identities
3 $\int_{0}^{2} \sqrt{4 - x^2} \, dx$
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Definite Integrals and the Fundamental Theorem
4 $\int\limits_{0}^{2\pi} \cos^2 x \, dx = k \int\limits_{0}^{\pi/2} \cos^2 x \, dx$
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Properties of Definite Integrals
5 $\frac{d}{dx}(f(x)) = \log x$
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Integration by Parts
5 $\frac{d}{dx}[f(x)] = ax + b$
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Integration as the Inverse of Differentiation
6 $\int 2^{x+2} \, dx$
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Integration as the Inverse of Differentiation
6 $\int\limits_{0}^{\frac{\pi}{6}} \sec^2\left(x - \frac{\pi}{6}\right) dx$
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Definite Integrals and the Fundamental Theorem
7 $\int \frac{2 \cos 2x - 1}{1 + 2 \sin x} \, dx$
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Integration using Trigonometric Identities
8 $\displaystyle\int e^{5 \log x}\, dx$
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Integration as the Inverse of Differentiation
9 $\displaystyle\int_0^a 3x^2\, dx = 8$
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Definite Integrals and the Fundamental Theorem
9 $\displaystyle\int_0^4 (e^{2x} + x)\, dx$
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Definite Integrals and the Fundamental Theorem
10 $\displaystyle\int \dfrac{\sec x}{\sec x - \tan x}\,dx$
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Integration using Trigonometric Identities
11 $\displaystyle\int_{-1}^{1} \dfrac{|x - 2|}{x - 2}\,dx$
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Definite Integrals and the Fundamental Theorem
11 $\displaystyle\int_{0}^{\pi/4} (\sin 2x)\,dx$
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Definite Integrals and the Fundamental Theorem
13 $\frac{\tan x - 1}{\tan x + 1}$
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Integration using Trigonometric Identities
14 $\frac{2}{1 + \cos 2x}$
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Integration using Trigonometric Identities
16 $\int e^{-x} \left( \frac{x+1}{x^2} \right) \, dx$
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Integration by Parts
17 $\int_{0}^{\pi/2} \log \tan x \, dx$
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Properties of Definite Integrals
In the following questions 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices :
20 $\displaystyle\int_{2}^{8} \dfrac{\sqrt{10 - x}}{\sqrt{x} + \sqrt{10 - x}}\,dx = 3$
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Properties of Definite Integrals
26 $\displaystyle\int \dfrac{2}{(1 - x)(1 + x^2)}\, dx$
Integration by Partial Fractions
26 $\int \frac{e^x}{\sqrt{e^{2x} - 4e^x - 5}} \, dx$
Integrals of Some Particular Functions
26 $\displaystyle\int \dfrac{x^2}{x^2 + 6x + 12}\, dx$
Integrals of Some Particular Functions
26 (a) $\int\limits_{0}^{\pi/4} \log (1 + \tan x) \, dx$
Properties of Definite Integrals
26 $\int\limits_{0}^{\frac{\pi}{2}} [\log (\sin x) - \log (2 \cos x)] \, dx$
Properties of Definite Integrals
26 $\displaystyle\int \dfrac{x^2 + x + 1}{(x+1)^2 (x+2)}\, dx$
Integration by Partial Fractions
27 $\int \frac{1}{\sqrt{x}\left(\sqrt{x} + 1\right)\left(\sqrt{x} + 2\right)} \, dx$
Integration by Partial Fractions
26 (b) $\int \frac{dx}{\sqrt{\sin^3 x \cos (x - \alpha)}}$
Integration by Substitution
27 (a) $\int \frac{x + 2}{\sqrt{x^2 - 4x - 5}} \, dx$
Integrals of Some Particular Functions
27 (a) $\displaystyle\int_{\pi/4}^{\pi/2} e^{2x}\left(\dfrac{1 - \sin 2x}{1 - \cos 2x}\right) dx$
Integrals of Special Forms
27 (a) $\displaystyle\int_{1/3}^{1} \dfrac{(x - x^3)^{1/3}}{x^4}\, dx$
Definite Integrals by Substitution
27 (b) $\int\limits_{-a}^{a} f(x) \, dx$
Properties of Definite Integrals
27 (b) $\displaystyle\int_{-2}^{2} \dfrac{x^2}{1 + 5^x}\, dx$
Properties of Definite Integrals
27 (b) $\displaystyle\int_{1}^{3} \left\{ |(x - 1)| + |(x - 2)| \right\} dx$
Properties of Definite Integrals
27 $\int_{1}^{3} \frac{\sqrt{4-x}}{\sqrt{x} + \sqrt{4-x}} \, dx$
Properties of Definite Integrals
27 $\int e^{\cot^{-1} x}\left(\frac{1 - x + x^2}{1 + x^2}\right) dx$
Integration by Parts
28 (a) $\displaystyle\int \dfrac{e^x}{\sqrt{5 - 4e^x - e^{2x}}}\, dx$
Integrals of Some Particular Functions
28 $\int \frac{x^2}{(x^2 + 4)(x^2 + 9)} \, dx$
Integration by Partial Fractions
28 (a) $\displaystyle\int_{0}^{2\pi} \dfrac{1}{1 + e^{\sin x}}\,dx$
Properties of Definite Integrals
28 $\int_{1}^{e} \frac{1}{\sqrt{4x^2 - (x \log x)^2}} \, dx$
Definite Integrals by Substitution
28 $\int\limits_{\log \sqrt{2}}^{\log \sqrt{3}} \frac{1}{(e^x + e^{-x})(e^x - e^{-x})} \, dx$
Definite Integrals by Substitution
28 (a) $\displaystyle\int_{-\pi/4}^{\pi/4} \dfrac{\cos 2x}{1 + \cos 2x}\,dx$
Properties of Definite Integrals
28 (b) $\displaystyle\int_0^{\pi/2} \sqrt{\sin x}\, \cos^5 x\, dx$
Definite Integrals by Substitution
28 (b) $\displaystyle\int \dfrac{x^4}{(x - 1)(x^2 + 1)}\,dx$
Integration by Partial Fractions
28 (b) $\displaystyle\int e^{x^2}(x^5 + 2x^3)\,dx$
Integration by Substitution
29 (a) $\int \frac{\cos x}{\sin 3x} \, dx$
Integration using Trigonometric Identities
28 $\int_{0}^{\pi/4} \log (1 + \tan x) \, dx$
Properties of Definite Integrals
29 (b) $\int x^2 \log (x^2 + 1) \, dx$
Integration by Parts
30 (a) $\int\limits_{-1}^{1} |x^4 - x| \, dx$
Properties of Definite Integrals
31 (a) $\int\limits_{0}^{\frac{\pi}{2}} e^x \sin x \, dx$
Integration by Parts
30 (b) $\int \frac{\sin^{-1} x}{(1 - x^2)^{3/2}} \, dx$
Integration by Parts
30 $\int\limits_{0}^{\frac{\pi}{2}} |\sin x - \cos x| \, dx$
Properties of Definite Integrals
30 $\int_{1}^{4} \frac{1}{\sqrt{2x+1} - \sqrt{2x-1}} \, dx$
Definite Integrals and the Fundamental Theorem
31 $\int_{-\pi/2}^{\pi/2} \frac{\sin^{100} x}{\sin^{100} x + \cos^{100} x} \, dx$
Properties of Definite Integrals
31 (b) $\int \frac{1}{\cos(x - a) \, \cos(x - b)} \, dx$
Integration using Trigonometric Identities
31 $\int e^x \left(\frac{1 - \sin x}{1 - \cos x}\right) dx$
Integrals of Special Forms
32 $\displaystyle\int_{0}^{\pi} \dfrac{x}{1 + \sin x}\,dx$
Properties of Definite Integrals
33 $\displaystyle\int_{0}^{\pi/2} \sin 2x \tan^{-1}(\sin x)\,dx$
Definite Integrals by Substitution

2022

33 q

1 $\int \dfrac{dx}{x^2 - 6x + 13}$
Integrals of Some Particular Functions
1 $\int \dfrac{dx}{\sqrt{4x - x^2}}$
Integrals of Some Particular Functions
2 (a) $\int \frac{\sin 3x}{\sin x} \, dx$
Integration using Trigonometric Identities
2 (b) $\int_{0}^{\frac{1}{2} \log 3} \frac{e^x}{e^{2x} + 1} \, dx$
Definite Integrals by Substitution
3 (a) $\frac{d}{dx}[F(x)] = \frac{\sec^4 x}{\operatorname{cosec}^4 x}$
Integration using Trigonometric Identities
3 (b) $\int \frac{\log x - 3}{(\log x)^4}\,dx$
Integration by Parts
6 (a) $\int \frac{\sqrt{\cot x}}{\sin x \cos x} \, dx$
Integration by Substitution
6 (b) $\int \frac{1}{x (x^2 + 4)} \, dx$
Integration by Substitution
6 $\displaystyle\int_0^1 x^2 e^x \, dx$
Integration by Parts
7 $\int_{0}^{2\pi} \dfrac{dx}{1 + e^{\sin x}}$
Properties of Definite Integrals
7 $\int_0^{\pi/4} \dfrac{dx}{1 + \tan x}$
Properties of Definite Integrals
8 (a) $\displaystyle\int \dfrac{1}{e^x + 1} \, dx$
Integration by Substitution
8 (a) $\int_{0}^{1} \tan^{-1} x \, dx$
Integration by Parts
8 (b) $\displaystyle\int_1^4 \{|x| + |3 - x|\} \, dx$
Definite Integrals and the Fundamental Theorem
8 $\int_{-1}^{2} |x^3 - x| \, dx$
Definite Integrals and the Fundamental Theorem
8 (b) $\int \frac{2x}{x^2 + 3x + 2} \, dx$
Integration by Partial Fractions
9 (a) $\int \frac{dx}{\sqrt{x} + \sqrt[3]{x}}$
Integration by Substitution
9 (a) $\int \frac{x^3 + x}{x^4 - 9}\,dx$
Integrals of Some Particular Functions
10 (a) $\int e^x \cdot \sin 2x\, dx$
Integration by Parts
9 (b) $\int_{-\pi}^{\pi} (3\sin x - 2)^2\,dx$
Properties of Definite Integrals
9 (b) $\int_0^{\pi/2} \frac{\cos x}{(1 + \sin x)(4 + \sin x)}\,dx$
Definite Integrals by Substitution
9 $\int_1^3 \dfrac{\sqrt{x}}{\sqrt{x} + \sqrt{4 - x}}\, dx$
Properties of Definite Integrals
10 $\int_{-\pi/2}^{\pi/2} (\sin |x| + \cos |x|) \, dx$
Properties of Definite Integrals
10 $\int_0^{\pi/2} \dfrac{1}{1 + (\tan x)^{2/3}}\, dx$
Properties of Definite Integrals
10 (b) $\int \dfrac{2x}{\left(x^2 + 1\right)\left(x^2 + 2\right)}\, dx$
Integration by Partial Fractions
11 $\displaystyle\int_{-3}^{3} \dfrac{x^4}{1 + e^x} \, dx$
Properties of Definite Integrals
12 $\int_{0}^{1} x (1 - x)^n \, dx$
Properties of Definite Integrals
12 (a) $\int \dfrac{x^2}{(x^2 + 1)(3x^2 + 4)} \, dx$
Integration by Partial Fractions
12 (b) $\int_{-2}^{1} \sqrt{5 - 4x - x^2} \, dx$
Integrals of Special Forms
12 $\displaystyle\int_0^{\pi} \dfrac{x}{9 \sin^2 x + 16 \cos^2 x} \, dx$
Properties of Definite Integrals
13 $\displaystyle\int_0^{\pi/2} (2 \log \cos x - \log \sin 2x) \, dx$
Properties of Definite Integrals
13 $\int_{0}^{\pi/2} \frac{x}{\sin x + \cos x} \, dx$
Properties of Definite Integrals
13 $\int_0^{\pi} \frac{x}{1 + \sin x}\,dx$
Properties of Definite Integrals

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 3 times: 2023, 2024, 2026* (* = numbers or wording changed that year)
27 (b) $\int \frac{x^2}{(x^2 + 9) \, (x^2 + 16)} \, dx$
Question 2
Asked 3 times: 2024*, 2025*, 2026 (* = numbers or wording changed that year)
9 $\displaystyle\int_{0}^{2a} \dfrac{1}{1 + 4x^2}\, dx = \dfrac{\pi}{6}$
Tap an option to check your answer.
Question 3
Asked 3 times: 2022*, 2024*, 2025 (* = numbers or wording changed that year)
27 (a) $\displaystyle\int \dfrac{2x}{(x^2 + 3)(x^2 - 5)}\ dx$
Question 4
Asked 2 times: 2023*, 2026 (* = numbers or wording changed that year)
27 $\int_{0}^{\pi} \dfrac{\sin^{2026} x}{\sin^{2026} x + \cos^{2026} x}\, dx$
Question 5
Asked 2 times: 2022*, 2026 (* = numbers or wording changed that year)
27 (b) $\displaystyle\int_{\frac{-\pi}{6}}^{\frac{\pi}{2}} (\sin |x| + \cos |x|)\, dx$
Question 6
Asked 2 times: 2022, 2026
26 (a) $\int \dfrac{dx}{x^{1/2} + x^{1/3}}$
Question 7
Asked 2 times: 2022, 2026
28 $\dfrac{d}{dx}(F(x)) = \dfrac{1}{e^x + 1}$
Question 8
Asked 2 times: 2022*, 2026 (* = numbers or wording changed that year)
28 (a) $\int \dfrac{\cos x}{(2 + \sin x)\,(4 + \sin x)}\, dx$
Question 9
Asked 2 times: 2025*, 2026 (* = numbers or wording changed that year)
27 (a) $\displaystyle\int \dfrac{x - \sin x}{1 - \cos x}\,dx$
Question 10
Asked 2 times: 2024*, 2026 (* = numbers or wording changed that year)
9 $\displaystyle\int \dfrac{dx}{\sqrt{25 - 16x^2}}$
Tap an option to check your answer.

7.2 Integration as an Inverse Process of Differentiation

Solved Examples

Exercise 7.1

7.3 Methods of Integration — Substitution

Solved Examples

Exercise 7.2

7.3 Methods of Integration — Trigonometric Identities

Solved Examples

Exercise 7.3

7.4 Integrals of Some Particular Functions

Solved Examples

Exercise 7.4

7.5 Integration by Partial Fractions

Solved Examples

Exercise 7.5

7.6 Integration by Parts

Solved Examples

Exercise 7.6

7.6 Integrals of Special Forms

Solved Examples

Exercise 7.7

7.7 Definite Integral and the Fundamental Theorem

Solved Examples

Exercise 7.8

7.9 Evaluation of Definite Integrals by Substitution

Solved Examples

Exercise 7.9

7.10 Some Properties of Definite Integrals

Solved Examples

Exercise 7.10

Miscellaneous Exercise on Chapter 7

Solved Examples

Miscellaneous Exercise

What the board has asked from this chapter

2026

2025

2024

2023

2022

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10