Mathematics · Textbook solutions

Indefinite Integration

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 292 questions

3.1 Elementary Integration

47 q

Solved Examples

Worked · 21

3.1 SolvedEx.1 $\int (x^3 + 3^x)\, dx$
3.1 SolvedEx.2 $\int \left( \sin x + \dfrac{1}{x} + \dfrac{1}{\sqrt[3]{x}} \right) dx$
3.1 SolvedEx.3 $\int (\tan x + \cot x)^2\, dx$
3.1 SolvedEx.4 $\int \dfrac{\sqrt{x} + 1}{x + \sqrt{x}}\, dx$
3.1 SolvedEx.5 $\int \dfrac{e^{4 \log x} - e^{5 \log x}}{x^5}\, dx$
3.1 SolvedEx.6 $\int \dfrac{2x + 3}{5x - 1}\, dx$
3.1 SolvedEx.7 $\int \dfrac{1}{\sqrt{3x + 1} - \sqrt{3x - 5}}\, dx$
3.1 SolvedEx.8 $\int \dfrac{2x - 7}{\sqrt{3x - 2}}\, dx$
3.1 SolvedEx.9 $\int \dfrac{x^3}{x - 1}\, dx$
3.1 SolvedEx.10 $\int \dfrac{3^x - 4^x}{5^x}\, dx$
3.1 SolvedEx.11 $\int \cos^3 x\, dx$
3.1 SolvedEx.12 $\int \sqrt{1 + \sin 3x}\, dx$
3.1 SolvedEx.13 $\int \sin^4 x\, dx$
3.1 SolvedEx.14 $\int \sin 5x \cdot \cos 7x\, dx$
3.1 SolvedEx.15 $\int \dfrac{\sin^3 x - \cos^3 x}{\sin^2 x \cdot \cos^2 x}\, dx$
3.1 SolvedEx.16 $\int \dfrac{1}{1 - \sin x}\, dx$
3.1 SolvedEx.17 $\int \left( \dfrac{\cos x}{1 - \cos x} \right) dx$
3.1 SolvedEx.18 $\int \dfrac{\cos x - \cos 2x}{1 - \cos x}\, dx$
3.1 SolvedEx.19 $\int \sin^{-1}(\cos 3x)\, dx$
3.1 SolvedEx.20 $\int \tan^{-1}\left( \dfrac{\sin 2x}{1 + \cos 2x} \right) dx$
3.1 SolvedEx.21 $\int \tan^{-1} \sqrt{\dfrac{1 - \sin x}{1 + \sin x}}\, dx$

Exercise 3.1

Practice · 26

Integrate the following functions w. r. t. x :
Ex 3.1 I (i) $\int (x^3 + x^2 - x + 1)\, dx$
Ex 3.1 I (ii) $\int x^2 \left(1 - \dfrac{2}{x}\right)^2 dx$
Ex 3.1 I (iii) $\int \left(3 \sec^2 x - \dfrac{4}{x} + \dfrac{1}{x\sqrt{x}} - 7\right) dx$
Ex 3.1 I (iv) $\int \left(2x^3 - 5x + \dfrac{3}{x} + \dfrac{4}{x^5}\right) dx$
Ex 3.1 I (v) $\int \dfrac{3x^3 - 2x + 5}{x\sqrt{x}}\, dx$
Evaluate :
Ex 3.1 II (i) $\int \tan^2 x\, dx$
Ex 3.1 II (ii) $\int \dfrac{\sin 2x}{\cos x}\, dx$
Ex 3.1 II (iii) $\int \dfrac{\sin x}{\cos^2 x}\, dx$
Ex 3.1 II (iv) $\int \dfrac{\cos 2x}{\sin^2 x}\, dx$
Ex 3.1 II (v) $\int \dfrac{\cos 2x}{\sin^2 x \cdot \cos^2 x}\, dx$
Ex 3.1 II (vi) $\int \dfrac{\sin x}{1 + \sin x}\, dx$
Ex 3.1 II (vii) $\int \dfrac{\tan x}{\sec x + \tan x}\, dx$
Ex 3.1 II (viii) $\int \sqrt{1 + \sin 2x}\, dx$
Ex 3.1 II (ix) $\int \sqrt{1 - \cos 2x}\, dx$
Ex 3.1 II (x) $\int \sin 4x \cdot \cos 3x\, dx$
Evaluate :
Ex 3.1 III (i) $\int \dfrac{x}{x + 2}\, dx$
Ex 3.1 III (ii) $\int \dfrac{4x + 3}{2x + 1}\, dx$
Ex 3.1 III (iii) $\int \dfrac{5x + 2}{3x - 4}\, dx$
Ex 3.1 III (iv) $\int \dfrac{x - 2}{\sqrt{x + 5}}\, dx$
Ex 3.1 III (v) $\int \dfrac{2x - 7}{\sqrt{4x - 1}}\, dx$
Ex 3.1 III (vi) $\int \dfrac{\sin 4x}{\cos 2x}\, dx$
Ex 3.1 III (vii) $\int \sqrt{1 + \sin 5x}\, dx$
Ex 3.1 III (viii) $\int \cos^2 x\, dx$
Ex 3.1 III (ix) $\int \dfrac{2}{\sqrt{x} - \sqrt{x + 3}}\, dx$
Ex 3.1 III (x) $\int \dfrac{3}{\sqrt{7x - 2} - \sqrt{7x - 5}}\, dx$
Ex 3.1 IV $f'(x) = x - \dfrac{3}{x^3}$

3.2 Methods of Integration — Substitution and Trigonometric

54 q

Solved Examples

Worked · 12

3.2A SolvedEx.1 $\int \frac{\cot (\log x)}{x}\, dx$
3.2A SolvedEx.2 $\int \frac{\cos \sqrt{x}}{\sqrt{x}}\, dx$
3.2A SolvedEx.3 $\int \frac{\sec^8 x}{\operatorname{cosec} x}\, dx$
3.2A SolvedEx.4 $\int \frac{1}{x + \sqrt{x}}\, dx$
3.2A SolvedEx.5 $\int 5^{5^x} \cdot 5^x\, dx$
3.2A SolvedEx.6 $\int \frac{1}{1 + e^{-x}}\, dx$
3.2A SolvedEx.7 $\int \frac{e^x (1 + x)}{\cos (x \cdot e^x)}\, dx$
3.2A SolvedEx.8 $\int \frac{1}{3x + 7x^{-n}}\, dx$
3.2A SolvedEx.9 $\int (3x + 2)\, \sqrt{x - 4}\, dx$
3.2A SolvedEx.10 $\int \frac{\sin (x + a)}{\cos (x - b)}\, dx$
3.2A SolvedEx.11 $\int \frac{e^x + 1}{e^x - 1}\, dx$
3.2A SolvedEx.12 $\int \frac{1}{1 - \tan x}\, dx$

Exercise 3.2 (A)

Practice · 42

Integrate the following functions w. r. t. x :
Ex 3.2(A) I (1) $\int \frac{(\log x)^n}{x}\, dx$
Ex 3.2(A) I (2) $\int \frac{(\sin^{-1} x)^{\frac{3}{2}}}{\sqrt{1 - x^2}}\, dx$
Ex 3.2(A) I (3) $\int \frac{1 + x}{x \cdot \sin (x + \log x)}\, dx$
Ex 3.2(A) I (4) $\int \frac{x \cdot \sec^2 (x^2)}{\sqrt{\tan^3 (x^2)}}\, dx$
Ex 3.2(A) I (5) $\int \frac{e^{3x}}{e^{3x} + 1}\, dx$
Ex 3.2(A) I (6) $\int \frac{(x^2 + 2)}{(x^2 + 1)} \cdot a^{x + \tan^{-1} x}\, dx$
Ex 3.2(A) I (7) $\int \frac{e^x \cdot \log (\sin e^x)}{\tan (e^x)}\, dx$
Ex 3.2(A) I (8) $\int \frac{e^{2x} + 1}{e^{2x} - 1}\, dx$
Ex 3.2(A) I (9) $\int \sin^4 x \cdot \cos^3 x\, dx$
Ex 3.2(A) I (10) $\int \frac{1}{4x + 5x^{-11}}\, dx$
Ex 3.2(A) I (11) $\int x^9 \cdot \sec^2 (x^{10})\, dx$
Ex 3.2(A) I (12) $\int e^{3 \log x} \cdot (x^4 + 1)^{-1}\, dx$
Ex 3.2(A) I (13) $\int \frac{\sqrt{\tan x}}{\sin x \cdot \cos x}\, dx$
Ex 3.2(A) I (14) $\int \frac{(x - 1)^2}{(x^2 + 1)^2}\, dx$
Ex 3.2(A) I (15) $\int \frac{2 \sin x \cdot \cos x}{3 \cos^2 x + 4 \sin^2 x}\, dx$
Ex 3.2(A) I (16) $\int \frac{1}{\sqrt{x} + \sqrt{x^3}}\, dx$
Ex 3.2(A) I (17) $\int \frac{10 x^9 + 10^x \cdot \log 10}{10^x + x^{10}}\, dx$
Ex 3.2(A) I (18) $\int \frac{x^{n-1}}{\sqrt{1 + 4x^n}}\, dx$
Ex 3.2(A) I (19) $\int (2x + 1)\sqrt{x + 2}\, dx$
Ex 3.2(A) I (20) $\int x^5 \cdot \sqrt{a^2 + x^2}\, dx$
Ex 3.2(A) I (21) $\int (5 - 3x)(2 - 3x)^{-\frac{1}{2}}\, dx$
Ex 3.2(A) I (22) $\int \frac{7 + 4x + 5x^2}{(2x + 3)^{\frac{3}{2}}}\, dx$
Ex 3.2(A) I (23) $\int \frac{x^2}{\sqrt{9 - x^6}}\, dx$
Ex 3.2(A) I (24) $\int \frac{1}{x (x^3 - 1)}\, dx$
Ex 3.2(A) I (25) $\int \frac{1}{x \cdot \log x \cdot \log (\log x)}\, dx$
Integrate the following functions w. r. t. x :
Ex 3.2(A) II (1) $\int \frac{\cos 3x - \cos 4x}{\sin 3x + \sin 4x}\, dx$
Ex 3.2(A) II (2) $\int \frac{\cos x}{\sin (x - a)}\, dx$
Ex 3.2(A) II (3) $\int \frac{\sin (x - a)}{\cos (x + b)}\, dx$
Ex 3.2(A) II (4) $\int \frac{1}{\sin x \cdot \cos x + 2 \cos^2 x}\, dx$
Ex 3.2(A) II (5) $\int \frac{\sin x + 2 \cos x}{3 \sin x + 4 \cos x}\, dx$
Ex 3.2(A) II (6) $\int \frac{1}{2 + 3 \tan x}\, dx$
Ex 3.2(A) II (7) $\int \frac{4 e^x - 25}{2 e^x - 5}\, dx$
Ex 3.2(A) II (8) $\int \frac{20 + 12 e^x}{3 e^x + 4}\, dx$
Ex 3.2(A) II (9) $\int \frac{3 e^{2x} + 5}{4 e^{2x} - 5}\, dx$
Ex 3.2(A) II (10) $\int \cos^8 x \cdot \cot x\, dx$
Ex 3.2(A) II (11) $\int \tan^5 x\, dx$
Ex 3.2(A) II (12) $\int \cos^7 x\, dx$
Ex 3.2(A) II (13) $\int \tan 3x \cdot \tan 2x \cdot \tan x\, dx$
Ex 3.2(A) II (14) $\int \sin^5 x \cdot \cos^8 x\, dx$
Ex 3.2(A) II (15) $\int 3^{\cos^2 x} \cdot \sin 2x\, dx$
Ex 3.2(A) II (16) $\int \frac{\sin 6x}{\sin 10x \cdot \sin 4x}\, dx$
Ex 3.2(A) II (17) $\int \frac{\sin x \cdot \cos^3 x}{1 + \cos^2 x}\, dx$

3.2 Special Integrals

45 q

Solved Examples

Worked · 16

3.2B SolvedEx.1 $\int \frac{1}{4x^2 + 11}\, dx$
3.2B SolvedEx.2 $\int \frac{1}{a^2 - b^2 x^2}\, dx$
3.2B SolvedEx.3 $\int \frac{1}{\sqrt{3x^2 - 7}}\, dx$
3.2B SolvedEx.4 $\int \frac{1}{x^2 + 8x + 12}\, dx$
3.2B SolvedEx.5 $\int \frac{1}{\sqrt{3x^2 - 4x + 2}}\, dx$
3.2B SolvedEx.6 $\int \frac{1}{3 - 10x - 25x^2}\, dx$
3.2B SolvedEx.7 $\int \frac{1}{\sqrt{1 + x - x^2}}\, dx$
3.2B SolvedEx.8 $\int \frac{\sin 2x}{3\sin^4 x - 4\sin^2 x + 1}\, dx$
3.2B SolvedEx.9 $\int \frac{e^{\frac{x}{2}}}{\sqrt{e^{-x} - e^{x}}}\, dx$
3.2B SolvedEx.10 $\int \left(\sqrt{\tan x} + \sqrt{\cot x}\right) dx$
3.2B SolvedEx.11 $\int \frac{1}{5 - 4\cos x}\, dx$
3.2B SolvedEx.12 $\int \frac{1}{2 - 3\sin 2x}\, dx$
3.2B SolvedEx.13 $\int \frac{1}{3 - 2\sin x + 5\cos x}\, dx$
3.2B SolvedEx.14 $\int \frac{1}{\sin x - \sqrt{3}\cos x}\, dx$
3.2B SolvedEx.15 $\int \frac{1}{3 + 2\sin^2 x + 5\cos^2 x}\, dx$
3.2B SolvedEx.16 $\int \frac{\cos\theta}{\cos 3\theta}\, d\theta$

Exercise 3.2 (B)

Practice · 29

Evaluate the following :
Ex 3.2(B) I (i) $\int \frac{1}{4x^2 - 3} \, dx$
Ex 3.2(B) I (ii) $\int \frac{1}{25 - 9x^2} \, dx$
Ex 3.2(B) I (iii) $\int \frac{1}{7 + 2x^2} \, dx$
Ex 3.2(B) I (iv) $\int \frac{1}{\sqrt{3x^2 + 8}} \, dx$
Ex 3.2(B) I (v) $\int \frac{1}{\sqrt{11 - 4x^2}} \, dx$
Ex 3.2(B) I (vi) $\int \frac{1}{\sqrt{2x^2 - 5}} \, dx$
Ex 3.2(B) I (vii) $\int \sqrt{\frac{9 + x}{9 - x}} \, dx$
Ex 3.2(B) I (viii) $\int \sqrt{\frac{2 + x}{2 - x}} \, dx$
Ex 3.2(B) I (ix) $\int \sqrt{\frac{10 + x}{10 - x}} \, dx$
Ex 3.2(B) I (x) $\int \frac{1}{x^2 + 8x + 12} \, dx$
Ex 3.2(B) I (xi) $\int \frac{1}{1 + x - x^2} \, dx$
Ex 3.2(B) I (xii) $\int \frac{1}{4x^2 - 20x + 17} \, dx$
Ex 3.2(B) I (xiii) $\int \frac{1}{5 - 4x - 3x^2} \, dx$
Ex 3.2(B) I (xiv) $\int \frac{1}{\sqrt{3x^2 + 5x + 7}} \, dx$
Ex 3.2(B) I (xv) $\int \frac{1}{\sqrt{x^2 + 8x - 20}} \, dx$
Ex 3.2(B) I (xvi) $\int \frac{1}{\sqrt{8 - 3x + 2x^2}} \, dx$
Ex 3.2(B) I (xvii) $\int \frac{1}{\sqrt{(x - 3)(x + 2)}} \, dx$
Ex 3.2(B) I (xviii) $\int \frac{1}{4 + 3\cos^2 x} \, dx$
Ex 3.2(B) I (xix) $\int \frac{1}{\cos 2x + 3\sin^2 x} \, dx$
Ex 3.2(B) I (xx) $\int \frac{\sin x}{\sin 3x} \, dx$
Integrate the following functions w. r. t. x :
Ex 3.2(B) II (i) $\int \frac{1}{3 + 2\sin x} \, dx$
Ex 3.2(B) II (ii) $\int \frac{1}{4 - 5\cos x} \, dx$
Ex 3.2(B) II (iii) $\int \frac{1}{2 + \cos x - \sin x} \, dx$
Ex 3.2(B) II (iv) $\int \frac{1}{3 + 2\sin x - \cos x} \, dx$
Ex 3.2(B) II (v) $\int \frac{1}{3 - 2\cos 2x} \, dx$
Ex 3.2(B) II (vi) $\int \frac{1}{2\sin 2x - 3} \, dx$
Ex 3.2(B) II (vii) $\int \frac{1}{3 + 2\sin 2x + 4\cos 2x} \, dx$
Ex 3.2(B) II (viii) $\int \frac{1}{\cos x - \sin x} \, dx$
Ex 3.2(B) II (ix) $\int \frac{1}{\cos x - \sqrt{3}\,\sin x} \, dx$

3.2 Integrals of the Type (px+q) over a Quadratic

12 q

Solved Examples

Worked · 3

3.2C SolvedEx.1 $\int \frac{2x-3}{3x^2+4x+5}\, dx$
3.2C SolvedEx.2 $\int \sqrt{\frac{x-5}{x-7}}\, dx$
3.2C SolvedEx.3 $\int \sqrt{\frac{8-x}{x}}\, dx$

Exercise 3.2 (C)

Practice · 9

Evaluate :
Ex 3.2(C) I (i) $\int \frac{3x+4}{x^2+6x+5}\, dx$
Ex 3.2(C) I (ii) $\int \frac{2x+1}{x^2+4x-5}\, dx$
Ex 3.2(C) I (iii) $\int \frac{2x+3}{2x^2+3x-1}\, dx$
Ex 3.2(C) I (iv) $\int \frac{3x+4}{\sqrt{2x^2+2x+1}}\, dx$
Ex 3.2(C) I (v) $\int \frac{7x+3}{\sqrt{3+2x-x^2}}\, dx$
Ex 3.2(C) I (vi) $\int \sqrt{\frac{x-7}{x-9}}\, dx$
Ex 3.2(C) I (vii) $\int \sqrt{\frac{9-x}{x}}\, dx$
Ex 3.2(C) I (viii) $\int \frac{3\cos x}{4\sin^2 x + 4\sin x - 1}\, dx$
Ex 3.2(C) I (ix) $\int \sqrt{\frac{e^{3x} - e^{2x}}{e^x + 1}}\, dx$

3.3 Integration by Parts

57 q

Solved Examples

Worked · 15

3.3 SolvedEx.1 $\int x^2\, 5^x\, dx$
3.3 SolvedEx.2 $\int x \tan^{-1} x\, dx$
3.3 SolvedEx.3 $\int \frac{x}{1 - \sin x}\, dx$
3.3 SolvedEx.4 $\int e^{2x} \sin 3x\, dx$
3.3 SolvedEx.5 $\int \left[ \log (\log x) + \frac{1}{(\log x)^2} \right] dx$
3.3 SolvedEx.6 $\int \sqrt{a^2 - x^2}\, dx$
3.3 SolvedEx.7 $\int \sqrt{a^2 + x^2}\, dx$
3.3 SolvedEx.8 $\int \sqrt{x^2 - a^2}\, dx$
3.3 SolvedEx.9 $\int x \sin^{-1} x\, dx$
3.3 SolvedEx.10 $\int \cos^{-1} \sqrt{x}\, dx$
3.3 SolvedEx.11 $\int \sqrt{4 + 3x - 2x^2}\, dx$
3.3.3 SolvedEx.1 $\int e^x \left( \frac{2 + \sin 2x}{1 + \cos 2x} \right) dx$
3.3.3 SolvedEx.2 $\int e^x \left[ \frac{x + 2}{(x + 3)^2} \right] dx$
3.3.3 SolvedEx.3 $\int e^{\tan^{-1} x} \left( \frac{1 + x + x^2}{1 + x^2} \right) dx$
3.3.3 SolvedEx.4 $\int \frac{(x^2 + 1) \cdot e^x}{(x + 1)^2}\, dx$

Exercise 3.3

Practice · 42

Evaluate the following :
Ex 3.3 I (i) $\int x^2 \cdot \log x\, dx$
Ex 3.3 I (ii) $\int x^2 \cdot \sin 3x\, dx$
Ex 3.3 I (iii) $\int x \cdot \tan^{-1} x\, dx$
Ex 3.3 I (iv) $\int x^2 \cdot \tan^{-1} x\, dx$
Ex 3.3 I (v) $\int x^3 \cdot \tan^{-1} x\, dx$
Ex 3.3 I (vi) $\int (\log x)^2\, dx$
Ex 3.3 I (vii) $\int \sec^3 x\, dx$
Ex 3.3 I (viii) $\int x \cdot \sin^2 x\, dx$
Ex 3.3 I (ix) $\int x^3 \cdot \log x\, dx$
Ex 3.3 I (x) $\int e^{2x} \cdot \cos 3x\, dx$
Ex 3.3 I (xi) $\int x \cdot \sin^{-1} x\, dx$
Ex 3.3 I (xii) $\int x^2 \cdot \cos^{-1} x\, dx$
Ex 3.3 I (xiii) $\int \frac{\log (\log x)}{x}\, dx$
Ex 3.3 I (xiv) $\int \frac{t \sin^{-1} t}{\sqrt{1 - t^2}}\, dt$
Ex 3.3 I (xv) $\int \cos \sqrt{x}\, dx$
Ex 3.3 I (xvi) $\int \sin \theta \cdot \log (\cos \theta)\, d\theta$
Ex 3.3 I (xvii) $\int x \cdot \cos^3 x\, dx$
Ex 3.3 I (xviii) $\int \frac{\sin (\log x)^2}{x} \cdot \log x\, dx$
Ex 3.3 I (xix) $\int \frac{\log x}{x}\, dx$
Ex 3.3 I (xx) $\int x \cdot \sin 2x \cdot \cos 5x\, dx$
Ex 3.3 I (xxi) $\int \cos \left(\sqrt[3]{x}\right)\, dx$
Integrate the following functions w. r. t. x :
Ex 3.3 II (i) $\int e^{2x} \cdot \sin 3x\, dx$
Ex 3.3 II (ii) $\int e^{-x} \cdot \cos 2x\, dx$
Ex 3.3 II (iii) $\int \sin (\log x)\, dx$
Ex 3.3 II (iv) $\int \sqrt{5x^2 + 3}\, dx$
Ex 3.3 II (v) $\int x^2 \cdot \sqrt{a^2 - x^6}\, dx$
Ex 3.3 II (vi) $\int \sqrt{(x - 3)(7 - x)}\, dx$
Ex 3.3 II (vii) $\int \sqrt{4^x (4^x + 4)}\, dx$
Ex 3.3 II (viii) $\int (x + 1)\sqrt{2x^2 + 3}\, dx$
Ex 3.3 II (ix) $\int x \sqrt{5 - 4x - x^2}\, dx$
Ex 3.3 II (x) $\int \sec^2 x \cdot \sqrt{\tan^2 x + \tan x - 7}\, dx$
Ex 3.3 II (xi) $\int \sqrt{x^2 + 2x + 5}\, dx$
Ex 3.3 II (xii) $\int \sqrt{2x^2 + 3x + 4}\, dx$
Integrate the following functions w. r. t. x :
Ex 3.3 III (i) $\int (2 + \cot x - \operatorname{cosec}^2 x) \cdot e^x\, dx$
Ex 3.3 III (ii) $\int \left(\frac{1 + \sin x}{1 + \cos x}\right) \cdot e^x\, dx$
Ex 3.3 III (iii) $\int e^x \cdot \left(\frac{1}{x} - \frac{1}{x^2}\right)\, dx$
Ex 3.3 III (iv) $\int \left(\frac{x}{(x + 1)^2}\right) \cdot e^x\, dx$
Ex 3.3 III (v) $\int \frac{e^x}{x} \left[x (\log x)^2 + 2 (\log x)\right]\, dx$
Ex 3.3 III (vi) $\int e^{5x} \cdot \left(\frac{5x \cdot \log x + 1}{x}\right)\, dx$
Ex 3.3 III (vii) $\int e^{\sin^{-1} x} \cdot \left(\frac{x + \sqrt{1 - x^2}}{\sqrt{1 - x^2}}\right)\, dx$
Ex 3.3 III (viii) $\int \log (1 + x)^{(1 + x)}\, dx$
Ex 3.3 III (ix) $\int \operatorname{cosec} (\log x) \left[1 - \cot (\log x)\right]\, dx$

3.4 Integration by Partial Fraction

28 q

Solved Examples

Worked · 5

3.4 SolvedEx.1 $\int \frac{3x^2 + 4x - 5}{(x^2 - 1)(x + 2)}\, dx$
3.4 SolvedEx.2 $\int \frac{2x^2 - 3}{(x^2 - 5)(x^2 + 4)}\, dx$
3.4 SolvedEx.3 $\int \frac{1}{(\sin \theta)(3 + 2 \cos \theta)}\, d\theta$
3.4 SolvedEx.4 $\int \frac{1}{2 \cos x + \sin 2x}\, dx$
3.4 SolvedEx.5 $\int \frac{\tan \theta + \tan^3 \theta}{1 + \tan^3 \theta}\, d\theta$

Exercise 3.4

Practice · 23

Integrate the following w. r. t. x :
Ex 3.4 I (i) $\int \frac{x^2 + 2}{(x - 1)(x + 2)(x + 3)} \, dx$
Ex 3.4 I (ii) $\int \frac{x^2}{(x^2 + 1)(x^2 - 2)(x^2 + 3)} \, dx$
Ex 3.4 I (iii) $\int \frac{12x + 3}{6x^2 + 13x - 63} \, dx$
Ex 3.4 I (iv) $\int \frac{2x}{4 - 3x - x^2} \, dx$
Ex 3.4 I (v) $\int \frac{x^2 + x - 1}{x^2 + x - 6} \, dx$
Ex 3.4 I (vi) $\int \frac{6x^3 + 5x^2 - 7}{3x^2 - 2x - 1} \, dx$
Ex 3.4 I (vii) $\int \frac{12x^2 - 2x - 9}{(4x^2 - 1)(x + 3)} \, dx$
Ex 3.4 I (viii) $\int \frac{1}{x(x^5 + 1)} \, dx$
Ex 3.4 I (ix) $\int \frac{2x^2 - 1}{x^4 + 9x^2 + 20} \, dx$
Ex 3.4 I (x) $\int \frac{x^2 + 3}{(x^2 - 1)(x^2 - 2)} \, dx$
Ex 3.4 I (xi) $\int \frac{2x}{(2 + x^2)(3 + x^2)} \, dx$
Ex 3.4 I (xii) $\int \frac{2^x}{4^x - 3 \cdot 2^x - 4} \, dx$
Ex 3.4 I (xiii) $\int \frac{3x - 2}{(x + 1)^2 (x + 3)} \, dx$
Ex 3.4 I (xiv) $\int \frac{5x^2 + 20x + 6}{x^3 + 2x^2 + x} \, dx$
Ex 3.4 I (xv) $\int \frac{1}{x(1 + 4x^3 + 3x^6)} \, dx$
Ex 3.4 I (xvi) $\int \frac{1}{x^3 - 1} \, dx$
Ex 3.4 I (xvii) $\int \frac{(3 \sin x - 2) \cdot \cos x}{5 - 4 \sin x - \cos^2 x} \, dx$
Ex 3.4 I (xviii) $\int \frac{1}{\sin x + \sin 2x} \, dx$
Ex 3.4 I (xix) $\int \frac{1}{2 \sin x + \sin 2x} \, dx$
Ex 3.4 I (xx) $\int \frac{1}{\sin 2x + \cos x} \, dx$
Ex 3.4 I (xxi) $\int \frac{1}{\sin x \cdot (3 + 2 \cos x)} \, dx$
Ex 3.4 I (xxii) $\int \frac{5 \cdot e^x}{(e^x + 1)(e^{2x} + 9)} \, dx$
Ex 3.4 I (xxiii) $\int \frac{2 \log x + 3}{x(3 \log x + 2)[(\log x)^2 + 1]} \, dx$

Miscellaneous Exercise 3

49 q

Choose the correct option

Practice · 20

Misc I Q.1 $\int \frac{1+x+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}}\, dx =$
Tap an option to check your answer.
Misc I Q.2 $\int \frac{1}{x+x^5}\, dx = f(x)+c,$
Tap an option to check your answer.
Misc I Q.3 $\int \frac{\log(3x)}{x\log(9x)}\, dx =$
Tap an option to check your answer.
Misc I Q.4 $\int \frac{\sin^m x}{\cos^{m+2}x}\, dx =$
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Misc I Q.5 $\int \tan(\sin^{-1}x)\, dx =$
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Misc I Q.6 $\int \frac{x-\sin x}{1-\cos x}\, dx =$
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Misc I Q.7 $f(x)=\frac{\sin^{-1}x}{\sqrt{1-x^2}},\ g(x)=e^{\sin^{-1}x},$
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Misc I Q.8 $\int \tan^3 x\sec^3 x\, dx = \left(\frac{1}{m}\right)\sec^m x - \left(\frac{1}{n}\right)\sec^n x + c,$
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Misc I Q.9 $\int \frac{1}{\cos x-\cos^2 x}\, dx =$
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Misc I Q.10 $\int \frac{\sqrt{\cot x}}{\sin x\,\cos x}\, dx =$
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Misc I Q.11 $\int \frac{e^x(x-1)}{x^2}\, dx =$
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Misc I Q.12 $\int \sin(\log x)\, dx =$
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Misc I Q.13 $\int x^x(1+\log x)\, dx =$
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Misc I Q.14 $\int \cos^{-\frac{3}{7}}x\,\sin^{-\frac{11}{7}}x\, dx =$
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Misc I Q.15 $2\int \frac{\cos^2 x-\sin^2 x}{\cos^2 x+\sin^2 x}\, dx =$
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Misc I Q.16 $\int \frac{dx}{\cos x\sqrt{\sin^2 x-\cos^2 x}} =$
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Misc I Q.17 $\int \frac{\log x}{(\log ex)^2}\, dx =$
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Misc I Q.18 $\int [\sin(\log x)+\cos(\log x)]\, dx =$
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Misc I Q.19 $\int \frac{\cos 2x-1}{\cos 2x+1}\, dx =$
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Misc I Q.20 $\int \frac{e^{2x}+e^{-2x}}{e^x}\, dx =$
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Integrate / Evaluate

Practice · 9

Misc II Q.1 $x$
Misc II Q.2 $x$
Misc II Q.3 $x$
Misc II Q.4 $t$
Misc II Q.5 $x$
Misc II Q.6 $\theta$
Misc II Q.7 $x$
Misc II Q.8 $x$
Misc II Q.9 $x$

Integrate the following

Practice · 20

Misc III Q.1 $\ \dfrac{(1+\log x)^3}{x}$
Misc III Q.2 $\ \cot^{-1}(1-x+x^2)$
Misc III Q.3 $\ \dfrac{1}{x\cdot\sin^2(\log x)}$
Misc III Q.4 $\ \sqrt{x}\,\sec\left(x^{\frac{3}{2}}\right)\cdot\tan\left(x^{\frac{3}{2}}\right)$
Misc III Q.5 $\ \log(1+\cos x) - x\cdot\tan\left(\dfrac{x}{2}\right)$
Misc III Q.6 $\ \dfrac{x^2}{\sqrt{1-x^6}}$
Misc III Q.7 $\ \dfrac{1}{(1-\cos 4x)(3-\cot 2x)}$
Misc III Q.8 $\ \log(\log x) + (\log x)^{-2}$
Misc III Q.9 $\ \dfrac{1}{2\cos x+3\sin x}$
Misc III Q.10 $\ \dfrac{1}{x^3\cdot\sqrt{x^2-1}}$
Misc III Q.11 $\ \dfrac{3x+1}{\sqrt{-2x^2+x+3}}$
Misc III Q.12 $\ \log(x^2+1)$
Misc III Q.13 $\ e^{2x}\cdot\sin x\cdot\cos x$
Misc III Q.14 $\ \dfrac{x^2}{(x-1)(3x-1)(3x-2)}$
Misc III Q.15 $\ \dfrac{1}{\sin x+\sin 2x}$
Misc III Q.16 $\ \sec^2 x\cdot\sqrt{7+2\tan x-\tan^2 x}$
Misc III Q.17 $\ \dfrac{x+5}{x^3+3x^2-x-3}$
Misc III Q.18 $\ \dfrac{1}{x\cdot(x^5+1)}$
Misc III Q.19 $\ \dfrac{\sqrt{\tan x}}{\sin x\cdot\cos x}$
Misc III Q.20 $\ \sec^4 x\cdot\csc^2 x$

What the board has asked from this chapter

Questions per paper · 14 papers · 2015 to 2026

June 2026

4 q

Q. 1. (vi) $f(x) = \frac{\sin^{-1} x}{\sqrt{1 - x^{2}}}$
Tap an option to check your answer.
Integration by Parts
Q. 11 $\int x \log x \ dx$
Integration by Parts
Q. 14 $\int \frac{e^{x}(1 + x)}{\cos^{2}(xe^{x})} dx$
Integration by Substitution
Q. 32 $\int \frac{2}{3 + 2\sin^{2} x + 5\cos^{2} x} dx$
Integrals of Trigonometric Functions

February 2026

4 q

Q. 2. (iii) $\int \frac{2x}{1 + x^{2}} dx$
Integration by Substitution
Q. 13 $\int \sqrt{1 + \sin 2x} \cdot dx$
Integrals of Trigonometric Functions
Q. 23 $\int \frac{\sin x}{\sin 3x} dx$
Integrals of Trigonometric Functions
Q. 31 $\int \frac{3x^{2} + 4x - 5}{(x^{2} - 1)(x + 2)} \cdot dx$
Integration by Partial Fractions

July 2025

4 q

Q. 2. (iv) $\int \frac{1}{\sqrt{x}} \, dx$
Elementary Integration and Standard Formulae
Q. 12 $\int \frac{dx}{x + x^{-10}}$
Integration by Substitution
Q. 22 $\int \frac{\sin(x + a)}{\cos(x - b)} \, dx$
Integrals of Trigonometric Functions
Q. 32 $\int \sqrt{a^{2} - x^{2}} \cdot dx = \frac{x}{2} \cdot \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \cdot \sin^{-1}\left(\frac{x}{a}\right) + c$
Special Integrals of Quadratic Forms

2025

4 q

Q. 1. vii. $\int x^{x}(1 + logx)dx$
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Elementary Integration and Standard Formulae
Q. 10 $\int x^{9}\sec^{2}\left( x^{10} \right)dx$
Integration by Substitution
Q. 11 $\int\frac{1}{25 - 9x^{2}}dx$
Special Integrals of Quadratic Forms
Q. 32 $\int e^{\sin^{- 1}x}\left( \frac{x + \sqrt{1 - x^{2}}}{\sqrt{1 - x^{2}}} \right)dx$
Integration by Parts

July 2024

4 q

Q. 2. (iii) $\int \frac{5^{x}}{3^{x}} dx$
Elementary Integration and Standard Formulae
Q. 10 $\int \frac{e^{x}(1 + x) dx}{\sin^{2}(xe^{x})}$
Integration by Substitution
Q. 23 $\int \frac{\sin(x + a)}{\cos(x - b)} dx$
Integrals of Trigonometric Functions
Q. 32 $\int x^{2} \sin 3x \, dx$
Integration by Parts

2024

2 q

Q. 33 $\int\frac{5e^{x}}{\left( e^{x} + 1 \right)\left( e^{2x} + 9 \right)}dx$
Integration by Partial Fractions
Q. 10 $\int logxdx$
Integration by Parts

March 2023

2 q

Q. 1. vi. $\int\cos^{3}x\ dx$
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Elementary Integration and Standard Formulae
Q. 23 $\int x\tan^{- 1}xdx$
Integration by Parts

March 2022

2 q

Q. 1. vii. $\int\frac{dx}{4x - 1} = Alog|4x - 1| + c$
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Elementary Integration and Standard Formulae
Q. 32 $u$
Integration by Parts

February 2020

1 q

Q. 10 $\int\frac{x}{x^{2} + 4x + 8}dx$
Integrals of the Type (px+q) over a Quadratic

March 2019

2 q

Q. 5 $\int\frac{dx}{9x^{2} + 1}$
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Special Integrals of Quadratic Forms
Q. 29 $\int\frac{dx}{\sqrt{x^{2} + a^{2}}} = log\left| x + \sqrt{x^{2} + a^{2}} \right| + c$
Special Integrals of Quadratic Forms

March 2018

2 q

Q. 4. (B) i. $\int e^{x}\left( \frac{cosx - sinx}{\sin^{2}x} \right)dx$
Integration by Parts
Q. 5. (B) ii. $\int\frac{dx}{a^{2} - x^{2}} = \frac{1}{2a}\log\left| \frac{a + x}{a - x} \right| + c$
Special Integrals of Quadratic Forms

March 2017

2 q

Q. 5. (A) ii. $u$
Integration by Parts
Q. 4. (B) iii. $\int\frac{\sin\sqrt{x}}{\sqrt{x}}dx$
Integration by Substitution

March 2016

1 q

Q. 6. (B) ii. $\int\sqrt{a^{2} - x^{2}}dx = \frac{x}{2}\sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2}\sin^{- 1}\left( \frac{x}{a} \right) + c$
Integration by Parts

March 2015

3 q

Q. 4. (B) i. $\int_{}^{}{e^{x}\left( \frac{\sqrt{1 - x^{2}}\sin^{- 1}x + 1}{\sqrt{1 - x^{2}}} \right)dx}$
Integration by Parts
Q. 5. (A) i. $\sec^{3}x$
Integration by Parts
Q. 5. (B) ii. $\int\sqrt{x^{2} - a^{2}}dx = \frac{x}{2}\sqrt{x^{2} - a^{2}} - \frac{a^{2}}{2}\log\left| x + \sqrt{x^{2} - a^{2}} \right| + c$
Integration by Parts

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 2 times: Jul 2024, Jun 2026* (* = numbers or wording changed that year)
Q. 14 $\int \frac{e^{x}(1 + x)}{\cos^{2}(xe^{x})} dx$
Question 2
Asked 2 times: Jul 2024, Jul 2025
Q. 22 $\int \frac{\sin(x + a)}{\cos(x - b)} \, dx$
Question 3
Asked 2 times: Mar 2016, Jul 2025
Q. 32 $\int \sqrt{a^{2} - x^{2}} \cdot dx = \frac{x}{2} \cdot \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \cdot \sin^{-1}\left(\frac{x}{a}\right) + c$
Question 4
Asked 2 times: Mar 2018, Mar 2024
Q. 5. (B) ii. $\int\frac{dx}{a^{2} - x^{2}} = \frac{1}{2a}\log\left| \frac{a + x}{a - x} \right| + c$
Question 5
Asked 2 times: Mar 2022, Mar 2024
Q. 10 $\int logxdx$
Question 6
Asked 2 times: Mar 2017, Mar 2022
Q. 32 $u$
Question 7
This type asked 6 times: Mar 2017, Jul 2024, Feb 2025, Jul 2025, Feb 2026, Jun 2026. This one: Feb 2026
Q. 2. (iii) $\int \frac{2x}{1 + x^{2}} dx$
Question 8
This type asked 6 times: Mar 2015, Mar 2016, Mar 2018, Mar 2019, Mar 2024, Jul 2025. This one: Mar 2019
Q. 29 $\int\frac{dx}{\sqrt{x^{2} + a^{2}}} = log\left| x + \sqrt{x^{2} + a^{2}} \right| + c$
Question 9
This type asked 5 times: Mar 2015, Mar 2023, Mar 2024, Jul 2024, Jun 2026. This one: Jun 2026
Q. 11 $\int x \log x \ dx$
Question 10
This type asked 4 times: Mar 2015, Mar 2018, Feb 2025, Jun 2026. This one: Jun 2026
Q. 1. (vi) $f(x) = \frac{\sin^{-1} x}{\sqrt{1 - x^{2}}}$
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3.1 Elementary Integration

Solved Examples

Exercise 3.1

3.2 Methods of Integration — Substitution and Trigonometric

Solved Examples

Exercise 3.2 (A)

3.2 Special Integrals

Solved Examples

Exercise 3.2 (B)

3.2 Integrals of the Type (px+q) over a Quadratic

Solved Examples

Exercise 3.2 (C)

3.3 Integration by Parts

Solved Examples

Exercise 3.3

3.4 Integration by Partial Fraction

Solved Examples

Exercise 3.4

Miscellaneous Exercise 3

Choose the correct option

Integrate / Evaluate

Integrate the following

What the board has asked from this chapter

June 2026

February 2026

July 2025

2025

July 2024

2024

March 2023

March 2022

February 2020

March 2019

March 2018

March 2017

March 2016

March 2015

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