Mathematics · Textbook solutions

Definite Integration

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 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) $\int_{2}^{3} \dfrac{dx}{x\,(x^{3} - 1)} =$
1. A. $\dfrac{1}{3} \log\left(\dfrac{208}{189}\right)$
2. B. $\dfrac{1}{3} \log\left(\dfrac{189}{208}\right)$
3. C. $\log\left(\dfrac{208}{189}\right)$
4. D. $\log\left(\dfrac{189}{208}\right)$
Misc 4 I (2) $\int_{0}^{\pi/2} \dfrac{\sin^{2} x \, dx}{(1 + \cos x)^{2}} =$
1. A. $\dfrac{4 - \pi}{2}$
2. B. $\dfrac{\pi - 4}{2}$
3. C. $4 - \dfrac{\pi}{2}$
4. D. $\dfrac{4 + \pi}{2}$
Misc 4 I (3) $\int_{0}^{\log 5} \dfrac{e^{x} \sqrt{e^{x} - 1}}{e^{x} + 3} \, dx =$
1. A. $3 + 2\pi$
2. B. $4 - \pi$
3. C. $2 + \pi$
4. D. $4 + \pi$
Misc 4 I (4) $\int_{0}^{\pi/2} \sin^{6} x \cos^{2} x \, dx =$
1. A. $\dfrac{7\pi}{256}$
2. B. $\dfrac{3\pi}{256}$
3. C. $\dfrac{5\pi}{256}$
4. D. $\dfrac{-5\pi}{256}$
Misc 4 I (5) $\int_{0}^{1} \dfrac{dx}{\sqrt{1 + x} - \sqrt{x}} = \dfrac{k}{3}$
1. A. $\sqrt{2}\,(2\sqrt{2} - 2)$
2. B. $\dfrac{\sqrt{2}}{3}\,(2 - 2\sqrt{2})$
3. C. $\dfrac{2\sqrt{2} - 2}{3}$
4. D. $4\sqrt{2}$
Misc 4 I (6) $\int_{1}^{2} \dfrac{1}{x^{2}}\, e^{\frac{1}{x}} \, dx =$
1. A. $\sqrt{e} + 1$
2. B. $\sqrt{e} - 1$
3. C. $\sqrt{e}\left(\sqrt{e} - 1\right)$
4. D. $\dfrac{\sqrt{e} - 1}{e}$
Misc 4 I (7) $\int_{2}^{e}\left[\dfrac{1}{\log x} - \dfrac{1}{(\log x)^{2}}\right] dx = a + \dfrac{b}{\log 2}$
1. A. $a = e,\ b = -2$
2. B. $a = e,\ b = 2$
3. C. $a = -e,\ b = 2$
4. D. $a = -e,\ b = -2$
Misc 4 I (8) $I_{1} = \int_{e}^{e^{2}} \dfrac{dx}{\log x}$
1. A. $I_{1} = \dfrac{1}{3} I_{2}$
2. B. $I_{1} + I_{2} = 0$
3. C. $I_{1} = 2 I_{2}$
4. D. $I_{1} = I_{2}$
Misc 4 I (9) $\int_{0}^{9} \dfrac{\sqrt{x}}{\sqrt{x} + \sqrt{9 - x}} \, dx =$
1. A. $9$
2. B. $\dfrac{9}{2}$
3. C. $0$
4. D. $1$
Misc 4 I (10) $\int_{-\pi/4}^{\pi/4} \log\left(\dfrac{2 + \sin\theta}{2 - \sin\theta}\right) d\theta$
1. A. $0$
2. B. $1$
3. C. $2$
4. D. $\pi$

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) $f(x) = a + bx + cx^{2}$