Mathematics · Textbook solutions

Definite Integration

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 112 questions

4.1 Definite Integral as Limit of a Sum

9 q

Solved Examples

Worked · 4

4.1 SolvedEx.1 $\int\limits_{1}^{2} (2x + 5)\ dx$
4.1 SolvedEx.2 $\int\limits_{2}^{3} 7^{x}\ dx$
4.1 SolvedEx.3 $\int\limits_{0}^{4} (x - x^{2})\ dx$
4.1 SolvedEx.4 $\int\limits_{0}^{\pi/2} \sin x\ dx$

Exercise 4.1

Practice · 5

Ex 4.1 I (1) $\int_{1}^{3} (3x - 4)\, dx$
Ex 4.1 I (2) $\int_{0}^{4} x^{2}\, dx$
Ex 4.1 I (3) $\int_{0}^{2} e^{x}\, dx$
Ex 4.1 I (4) $\int_{0}^{2} (3x^{2} - 1)\, dx$
Ex 4.1 I (5) $\int_{1}^{3} x^{3}\, dx$

4.2 Fundamental Theorem and Properties

70 q

Illustrations and Properties

Worked · 10

4.2 Ex.1 $\int_{2}^{5}(x^{2} - x)\,dx$
4.2.1 PropertyII Ex.1 $\int_{1}^{3} x\,dx$
4.2.1 PropertyII Ex.2 $\int_{3}^{1} x\,dx$
4.2.1 PropertyIII Ex.1 $\int_{\pi/6}^{\pi/3}\cos x\,dx$
4.2.1 PropertyIII Ex.2 $\int_{\pi/6}^{\pi/3}\cos t\,dt$
4.2.1 PropertyIV Ex.1 $\int_{-1}^{5}(2x + 3)\,dx = \int_{-1}^{3}(2x + 3)\,dx + \int_{3}^{5}(2x + 3)\,dx$
4.2.1 PropertyV Ex.1 $\int_{\pi/6}^{\pi/3}\sin^{2} x\,dx$
4.2.1 PropertyVI Ex.1 $\int_{0}^{\pi/4}\log(1 + \tan x)\,dx$
4.2.1 PropertyVIII Ex.1 $\int_{-\pi/4}^{\pi/4} x^{3}\sin^{4} x\,dx$
4.2.1 PropertyVIII Ex.2 $\int_{-1}^{1}\dfrac{x^{2}}{1 + x^{2}}\,dx$

Solved Examples

Worked · 15

4.2 SolvedEx.1 $\int_{1}^{3} \dfrac{1}{\sqrt{2+x} + \sqrt{x}}\ dx$
4.2 SolvedEx.2 $\int_{0}^{\pi/2}\sqrt{1-\cos 4x}\ dx$
4.2 SolvedEx.3 $\int_{0}^{\pi/2}\cos^{3}x\ dx$
4.2 SolvedEx.4 $\int_{0}^{\pi/4}\dfrac{\sec^{2}x}{2\tan^{2}x+5\tan x+1}\ dx$
4.2 SolvedEx.5 $\int_{1}^{2}\dfrac{\log x}{x^{2}}\ dx$
4.2 SolvedEx.6 $\int_{0}^{\pi/2}\dfrac{\cos x}{1+\cos x+\sin x}\ dx$
4.2 SolvedEx.7 $\int_{0}^{1/2}\dfrac{1}{(1-2x^{2})\sqrt{1-x^{2}}}\ dx$
4.2 SolvedEx.8 $\int_{0}^{2}\dfrac{2^{x}}{2^{x}\left(1+4^{x}\right)}\ dx$
4.2 SolvedEx.9 $\int_{-1}^{1}\left|\,5x-3\,\right|\ dx$
4.2 SolvedEx.10 $\int_{0}^{\pi/2}\dfrac{1}{1+\sqrt[3]{\tan x}}\ dx$
4.2 SolvedEx.11 $\int_{3}^{8}\dfrac{(11-x)^{2}}{x^{2}+(1-x)^{2}}\ dx$
4.2 SolvedEx.12 $\int_{0}^{\pi}x\,\sin^{2}x\ dx$
4.2 SolvedEx.13 $\int_{0}^{\pi}\cos^{2}x\ dx$
4.2 SolvedEx.14 $\int_{-\pi}^{\pi}\dfrac{x\,(1+\sin x)}{1+\cos^{2}x}\ dx$
4.2 SolvedEx.15 $\int_{0}^{3}x\,[\,x\,]\ dx$

Exercise 4.2

Practice · 45

Ex 4.2 I (1) $\int_{1}^{9} \dfrac{x + 1}{\sqrt{x}} \, dx$
Ex 4.2 I (2) $\int_{2}^{3} \dfrac{1}{x^{2} + 5x + 6} \, dx$
Ex 4.2 I (3) $\int_{0}^{\pi/4} \cot^{2} x$
Ex 4.2 I (4) $\int_{-\pi/4}^{\pi/4} \dfrac{1}{1 - \sin x} \, dx$
Ex 4.2 I (5) $\int_{3}^{5} \dfrac{1}{\sqrt{2x + 3} - \sqrt{2x - 3}} \, dx$
Ex 4.2 I (6) $\int_{0}^{1} \dfrac{x^{2} - 2}{x^{2} + 1} \, dx$
Ex 4.2 I (7) $\int_{0}^{\pi/4} \sin 4x \, \sin 3x \, dx$
Ex 4.2 I (8) $\int_{0}^{\pi/4} \sqrt{1 + \sin 2x} \, dx$
Ex 4.2 I (9) $\int_{0}^{\pi/4} \sin^{4} x \, dx$
Ex 4.2 I (10) $\int_{-4}^{2} \dfrac{1}{x^{2} + 4x + 13} \, dx$
Ex 4.2 I (11) $\int_{0}^{4} \dfrac{1}{\sqrt{4x - x^{2}}} \, dx$
Ex 4.2 I (12) $\int_{0}^{1} \dfrac{1}{\sqrt{3 + 2x - x^{2}}} \, dx$
Ex 4.2 I (13) $\int_{0}^{\pi/2} x \sin x \, dx$
Ex 4.2 I (14) $\int_{0}^{1} x \tan^{-1} x \, dx$
Ex 4.2 I (15) $\int_{0}^{\infty} x \, e^{-x} \, dx$
Ex 4.2 II (1) $\int_{0}^{\frac{1}{\sqrt{2}}} \dfrac{\sin^{-1} x}{(1 - x^{2})^{\frac{3}{2}}} \, dx$
Ex 4.2 II (2) $\int_{0}^{\pi/4} \dfrac{\sec^{2} x}{3 \tan^{2} x + 4 \tan x + 1} \, dx$
Ex 4.2 II (3) $\int_{0}^{\pi/4} \dfrac{\sin 2x}{\sin^{4} x + \cos^{4} x} \, dx$
Ex 4.2 II (4) $\int_{0}^{\pi/2} \sqrt{\cos x} \, \sin^{3} x \, dx$
Ex 4.2 II (5) $\int_{0}^{\pi/2} \dfrac{1}{5 + 4 \cos x} \, dx$
Ex 4.2 II (6) $\int_{0}^{\pi/4} \dfrac{\cos x}{4 - \sin^{2} x} \, dx$
Ex 4.2 II (7) $\int_{0}^{\pi/2} \dfrac{\cos x}{(1 + \sin x) \, (2 + \sin x)} \, dx$
Ex 4.2 II (8) $\int_{-1}^{1} \dfrac{1}{a^{2} e^{x} + b^{2} e^{-x}} \, dx$
Ex 4.2 II (9) $\int_{0}^{\pi} \dfrac{1}{3 + 2 \sin x + \cos x} \, dx$
Ex 4.2 II (10) $\int_{0}^{\pi/4} \sec^{4} x \, dx$
Ex 4.2 II (11) $\int_{0}^{1} \sqrt{\dfrac{1 - x}{1 + x}} \, dx$
Ex 4.2 II (12) $\int_{0}^{\pi} \sin^{3} x \, (1 + 2 \cos x) \, (1 + \cos x)^{2} \, dx$
Ex 4.2 II (13) $\int_{0}^{\pi/2} \sin 2x \, \tan^{-1}(\sin x) \, dx$
Ex 4.2 II (14) $\int_{\frac{1}{\sqrt{2}}}^{1} \dfrac{(e^{\cos^{-1} x}) \, (\sin^{-1} x)}{\sqrt{1 - x^{2}}} \, dx$
Ex 4.2 II (15) $\int_{2}^{3} \dfrac{\cos(\log x)}{x} \cdot dx$
Ex 4.2 III (1) $\int_{0}^{a} \dfrac{1}{x + \sqrt{a^{2} - x^{2}}} \, dx$
Ex 4.2 III (2) $\int_{0}^{\pi/2} \log \tan x \, dx$
Ex 4.2 III (3) $\int_{0}^{1} \log\left(\dfrac{1}{x} - 1\right) dx$
Ex 4.2 III (4) $\int_{0}^{\pi/2} \dfrac{\sin x - \cos x}{1 + \sin x \cdot \cos x} \, dx$
Ex 4.2 III (5) $\int_{0}^{3} x^{2} \, (3 - x)^{\frac{5}{2}} \, dx$
Ex 4.2 III (6) $\int_{-3}^{3} \dfrac{x^{3}}{9 - x^{2}} \, dx$
Ex 4.2 III (7) $\int_{-\pi/2}^{\pi/2} \log\left(\dfrac{2 + \sin x}{2 - \sin x}\right) dx$
Ex 4.2 III (8) $\int_{-\pi/4}^{\pi/4} \dfrac{x + \dfrac{\pi}{4}}{2 - \cos 2x} \, dx$
Ex 4.2 III (9) $\int_{-\pi/4}^{\pi/4} x^{3} \, \sin^{4} x \, dx$
Ex 4.2 III (10) $\int_{0}^{1} \dfrac{\log (x + 1)}{x^{2} + 1} \, dx$
Ex 4.2 III (11) $\int_{-1}^{1} \dfrac{x^{3} + 2}{\sqrt{x^{2} + 4}} \, dx$
Ex 4.2 III (12) $\int_{-a}^{a} \dfrac{x + x^{3}}{16 - x^{2}} \, dx$
Ex 4.2 III (13) $\int_{0}^{1} t^{2} \sqrt{1 - t} \, dt$
Ex 4.2 III (14) $\int_{0}^{\pi} x \, \sin x \, \cos^{2} x \, dx$
Ex 4.2 III (15) $\int_{0}^{1} \dfrac{\log x}{\sqrt{1 - x^{2}}} \, dx$

Miscellaneous Exercise 4

33 q

Choose the correct option

Practice · 10

Misc 4 I (1) $\int_{2}^{3} \dfrac{dx}{x\,(x^{3} - 1)} =$
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Misc 4 I (2) $\int_{0}^{\pi/2} \dfrac{\sin^{2} x \, dx}{(1 + \cos x)^{2}} =$
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Misc 4 I (3) $\int_{0}^{\log 5} \dfrac{e^{x} \sqrt{e^{x} - 1}}{e^{x} + 3} \, dx =$
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Misc 4 I (4) $\int_{0}^{\pi/2} \sin^{6} x \cos^{2} x \, dx =$
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Misc 4 I (5) $\int_{0}^{1} \dfrac{dx}{\sqrt{1 + x} - \sqrt{x}} = \dfrac{k}{3}$
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Misc 4 I (6) $\int_{1}^{2} \dfrac{1}{x^{2}}\, e^{\frac{1}{x}} \, dx =$
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Misc 4 I (7) $\int_{2}^{e}\left[\dfrac{1}{\log x} - \dfrac{1}{(\log x)^{2}}\right] dx = a + \dfrac{b}{\log 2}$
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Misc 4 I (8) $I_{1} = \int_{e}^{e^{2}} \dfrac{dx}{\log x}$
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Misc 4 I (9) $\int_{0}^{9} \dfrac{\sqrt{x}}{\sqrt{x} + \sqrt{9 - x}} \, dx =$
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Misc 4 I (10) $\int_{-\pi/4}^{\pi/4} \log\left(\dfrac{2 + \sin\theta}{2 - \sin\theta}\right) d\theta$
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Evaluate the following

Practice · 10

Misc 4 II (1) $\int_{0}^{\pi/2} \dfrac{\cos x}{3 \cos x + \sin x} \, dx$
Misc 4 II (2) $\int_{\pi/4}^{\pi/2} \dfrac{\cos\theta}{\left[\cos\dfrac{\theta}{2} + \sin\dfrac{\theta}{2}\right]^{3}} \, d\theta$
Misc 4 II (3) $\int_{0}^{1} \dfrac{1}{1 + \sqrt{x}} \, dx$
Misc 4 II (4) $\int_{0}^{\pi/4} \dfrac{\tan^{3} x}{1 + \cos 2x} \, dx$
Misc 4 II (5) $\int_{0}^{1} t^{5} \sqrt{1 - t^{2}} \, dt$
Misc 4 II (6) $\int_{0}^{1} (\cos^{-1} x)^{2} \, dx$
Misc 4 II (7) $\int_{-1}^{1} \dfrac{1 + x^{3}}{9 - x^{2}} \, dx$
Misc 4 II (8) $\int_{0}^{\pi} x \sin x \cos^{4} x \, dx$
Misc 4 II (9) $\int_{0}^{\pi} \dfrac{x}{1 + \sin^{2} x} \, dx$
Misc 4 II (10) $\int_{1}^{\infty} \dfrac{1}{\sqrt{x}\,(1 + x)} \, dx$

Evaluate

Practice · 10

Misc 4 III (1) $\int_{0}^{1} \left(\dfrac{1}{1 + x^{2}}\right) \sin^{-1}\left(\dfrac{2x}{1 + x^{2}}\right) dx$
Misc 4 III (2) $\int_{0}^{\pi/2} \dfrac{1}{6 - \cos x} \, dx$
Misc 4 III (3) $\int_{0}^{a} \dfrac{1}{a^{2} + ax - x^{2}} \, dx$
Misc 4 III (4) $\int_{\pi/5}^{3\pi/10} \dfrac{\sin x}{\sin x + \cos x} \, dx$
Misc 4 III (5) $\int_{0}^{1} \sin^{-1}\left(\dfrac{2x}{1 + x^{2}}\right) dx$
Misc 4 III (6) $\int_{0}^{\pi/4} \dfrac{\cos 2x}{1 + \cos 2x + \sin 2x} \, dx$
Misc 4 III (7) $\int_{0}^{\pi/2} \left(2 \log \sin x - \log \sin 2x\right) dx$
Misc 4 III (8) $\int_{0}^{\pi} \left(\sin^{-1} x + \cos^{-1} x\right)^{3} \sin^{3} x \, dx$
Misc 4 III (9) $\int_{0}^{4} \left[\sqrt{x^{2} + 2x + 3}\right]^{-1} dx$
Misc 4 III (10) $\int_{-2}^{3} \left|x - 2\right| dx$

Evaluate the following

Practice · 3

Misc 4 IV (1) $\int_{0}^{a} \sqrt{x} \, dx = 2a \int_{0}^{\pi/2} \sin^{3} x \, dx$
Misc 4 IV (2) $\int_{0}^{k} \dfrac{1}{2 + 8x^{2}} \, dx = \dfrac{\pi}{16}$
Misc 4 IV (3) $f(x) = a + bx + cx^{2}$

What the board has asked from this chapter

Questions per paper · 9 papers · 2016 to 2026

June 2026

2 q

Q. 10 $\int_{-1}^{1} |x| \, dx$
Properties of Definite Integrals
Q. 33 $\int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) dx$
Properties of Definite Integrals

February 2026

2 q

Q. 1. (vii) $\int_{1}^{2} \frac{1}{x^{2}} \cdot e^{\frac{1}{x}} dx =$
Tap an option to check your answer.
Methods of Evaluation of Definite Integrals
Q. 32 $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a + b - x) \cdot dx$
Properties of Definite Integrals

July 2025

2 q

Q. 13 $\int_{0}^{-1} e^{-x} \, dx$
Fundamental Theorem of Integral Calculus
Q. 33 $\int_{0}^{\frac{1}{2}} \frac{dx}{(1 - 2x^{2}) \cdot \sqrt{1 - x^{2}}}$
Methods of Evaluation of Definite Integrals

2025

2 q

Q. 12 $\int_{- \pi/4}^{\pi/4}\frac{1}{1 - sinx}dx$
Fundamental Theorem of Integral Calculus
Q. 33 $\int_{a}^{b}f(x)dx = \int_{a}^{b}f(a + b - x)dx$
Properties of Definite Integrals

July 2024

1 q

Q. 33 $\int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) \, dx$
Properties of Definite Integrals

2024

3 q

Q. 1. vi. $\int_{- \pi/4}^{\pi/4}x^{3}\sin^{4}xdx = k$
Tap an option to check your answer.
Properties of Definite Integrals
Q. 11 $\int_{0}^{\frac{\pi}{2}}\cos^{2}xdx$
Fundamental Theorem of Integral Calculus
Q. 34 $\int_{0}^{2a}f(x)dx = \int_{0}^{a}f(x)dx + \int_{0}^{a}f(2a - x)dx$
Properties of Definite Integrals

March 2022

1 q

Q. 34 $\int_{0}^{\pi}\log(1 + tanx)dx = \frac{\pi}{8}log2$
Properties of Definite Integrals

March 2018

1 q

Q. 4. (A) ii. $\int_{0}^{k}\frac{1}{2 + 8x^{2}}dx = \ \frac{\pi}{16}$
Tap an option to check your answer.
Methods of Evaluation of Definite Integrals

March 2016

1 q

Q. 6. (A) i. $\int_{0}^{2a}f(x)dx = \int_{0}^{a}f(x)dx + \int_{0}^{a}f(2a - x)dx$
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 2 times: Jul 2024, Jun 2026
Q. 33 $\int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) dx$
Question 2
Asked 2 times: Feb 2025, Feb 2026
Q. 32 $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a + b - x) \cdot dx$
Question 3
Asked 2 times: Mar 2016, Mar 2024
Q. 34 $\int_{0}^{2a}f(x)dx = \int_{0}^{a}f(x)dx + \int_{0}^{a}f(2a - x)dx$
Question 4
Asked once: Jun 2026
Q. 10 $\int_{-1}^{1} |x| \, dx$
Question 5
Asked once: Feb 2026
Q. 1. (vii) $\int_{1}^{2} \frac{1}{x^{2}} \cdot e^{\frac{1}{x}} dx =$
Tap an option to check your answer.
Question 6
Asked once: Jul 2025
Q. 13 $\int_{0}^{-1} e^{-x} \, dx$
Question 7
Asked once: Jul 2025
Q. 33 $\int_{0}^{\frac{1}{2}} \frac{dx}{(1 - 2x^{2}) \cdot \sqrt{1 - x^{2}}}$
Question 8
Asked once: Feb 2025
Q. 12 $\int_{- \pi/4}^{\pi/4}\frac{1}{1 - sinx}dx$
Question 9
Asked once: Mar 2024
Q. 11 $\int_{0}^{\frac{\pi}{2}}\cos^{2}xdx$
Question 10
Asked once: Mar 2024
Q. 1. vi. $\int_{- \pi/4}^{\pi/4}x^{3}\sin^{4}xdx = k$
Tap an option to check your answer.

4.1 Definite Integral as Limit of a Sum

Solved Examples

Exercise 4.1

4.2 Fundamental Theorem and Properties

Illustrations and Properties

Solved Examples

Exercise 4.2

Miscellaneous Exercise 4

Choose the correct option

Evaluate the following

Evaluate

Evaluate the following

What the board has asked from this chapter

June 2026

February 2026

July 2025

2025

July 2024

2024

March 2022

March 2018

March 2016

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