Integrals Class 12

Find anti derivative of sin 2x

Find anti derivative of cos 3x

Find anti derivative of e2x

Find an anti derivative (or integral) of the following functions by the method of inspection.
(ax + b)2

Find anti derivative of sin 2x - 4 e^3x

$\int\left(4 e^{3 x}+1\right) d x$

$\int x^2\left(1-\frac{1}{x^2}\right) d x$

$\int\left(a x^2+b x+c\right) d x$

$\int\left(2 x^2+e^x\right) d x$

$\int\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^2 d x$

$\int \frac{x^3+5 x^2-4}{x^2} d x$

$\int \frac{x^3+3 x+4}{\sqrt{x}} d x$

$\int \frac{x^3-x^2+x-1}{x-1} d x$

$\int(1-x) \sqrt{x} d x$

$\int \sqrt{x}\left(3 x^2+2 x+3\right) d x$

$\int\left(2 x-3 \cos x+e^x\right) d x$

$\int\left(2 x^2-3 \sin x+5 \sqrt{x}\right) d x$

$\int \sec x(\sec x+\tan x) d x$

$\int \frac{\sec ^2 x}{\operatorname{cosec}^2 x} d x$

$\int \frac{2-3 \sin x}{\cos ^2 x} d x$.

The anti derivative of $\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)$ equals
(A) $\frac{1}{3} x^{\frac{1}{3}}+2 x^{\frac{1}{2}}+\mathrm{C}$
(B) $\frac{2}{3} x^{\frac{2}{3}}+\frac{1}{2} x^2+\mathrm{C}$
(C) $\frac{2}{3} x^{\frac{3}{2}}+2 x^{\frac{1}{2}}+\mathrm{C}$
(D) $\frac{3}{2} x^{\frac{3}{2}}+\frac{1}{2} x^{\frac{1}{2}}+\mathrm{C}$

If $\frac{d}{d x} f(x)=4 x^3-\frac{3}{x^4}$ such that $f(2)=0$. Then $f(x)$ is
(A) $x^4+\frac{1}{x^3}-\frac{129}{8}$
(B) $x^3+\frac{1}{x^4}+\frac{129}{8}$
(C) $x^4+\frac{1}{x^3}+\frac{129}{8}$
(D) $x^3+\frac{1}{x^4}-\frac{129}{8}$

$\frac{2 x}{1+x^2}$

$\frac{(\log x)^2}{x}$

$\frac{1}{x+x \log x}$

$\sin x \sin (\cos x)$

$\sin (a x+b) \cos (a x+b)$

$\sqrt{a x+b}$

$x \sqrt{x+2}$

$x \sqrt{1+2 x^2}$

$(4 x+2) \sqrt{x^2+x+1}$

$\frac{1}{x-\sqrt{x}}$

$\frac{x}{\sqrt{x+4}}, x>0$

$\left(x^3-1\right)^{\frac{1}{3}} x^5$

$\frac{x^2}{\left(2+3 x^3\right)^3}$

$\frac{1}{x(\log x)^m}, x>0, m \neq 1$

$\frac{x}{9-4 x^2}$

$e^{2 x+3}$

$\frac{x}{e^{x^2}}$

$\frac{e^{\tan ^{-1} x}}{1+x^2}$

$\frac{e^{2 x}-1}{e^{2 x}+1}$

$\frac{e^{2 x}-e^{-2 x}}{e^{2 x}+e^{-2 x}}$

$\tan ^2(2 x-3)$

$\sec ^2(7-4 x)$

$\frac{\sin ^{-1} x}{\sqrt{1-x^2}}$

$\frac{2 \cos x-3 \sin x}{6 \cos x+4 \sin x}$

$\frac{1}{\cos ^2 x(1-\tan x)^2}$

$\frac{\cos \sqrt{x}}{\sqrt{x}}$

$\sqrt{\sin 2 x} \cos 2 x$

$\frac{\cos x}{\sqrt{1+\sin x}}$

$\cot x \log \sin x$

$\frac{\sin x}{1+\cos x}$

$\frac{\sin x}{(1+\cos x)^2}$

$\frac{1}{1+\cot x}$

$\frac{1}{1-\tan x}$

$\frac{\sqrt{\tan x}}{\sin x \cos x}$

$\frac{(1+\log x)^2}{x}$

$\frac{(x+1)(x+\log x)^2}{x}$

$\frac{x^3 \sin \left(\tan ^{-1} x^4\right)}{1+x^8}$

$\int \frac{10 x^9+10^x \log _e 10 d x}{x^{10}+10^x}$ equals
(A) $10^x-x^{10}+\mathrm{C}$
(B) $10^x+x^{10}+\mathrm{C}$
(C) $\left(10^x-x^{10}\right)^{-1}+\mathrm{C}$
(D) $\log \left(10^x+x^{10}\right)+\mathrm{C}$

$\int \frac{d x}{\sin ^2 x \cos ^2 x}$ equals
(A) $\tan x+\cot x+\mathrm{C}$
(B) $\tan x-\cot x+\mathrm{C}$
(C) $\tan x \cot x+\mathrm{C}$
(D) $\tan x-\cot 2 x+\mathrm{C}$

$\sin ^2(2 x+5)$

$\sin 3 x \cos 4 x$

$\cos 2 x \cos 4 x \cos 6 x$

$\sin ^3(2 x+1)$

$\sin ^3 x \cos ^3 x$

$\sin x \sin 2 x \sin 3 x$

$\sin 4 x \sin 8 x$

$\frac{1-\cos x}{1+\cos x}$

$\frac{\cos x}{1+\cos x}$

$\sin ^4 x$

$\cos ^4 2 x$

$\frac{\sin ^2 x}{1+\cos x}$

$\frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha}$

$\frac{\cos x-\sin x}{1+\sin 2 x}$

$\tan ^3 2 x \sec 2 x$

$\tan ^4 x$

$\frac{\sin ^3 x+\cos ^3 x}{\sin ^2 x \cos ^2 x}$

$\frac{\cos 2 x+2 \sin ^2 x}{\cos ^2 x}$

$\frac{1}{\sin x \cos ^3 x}$

$\frac{\cos 2 x}{(\cos x+\sin x)^2}$

$\sin ^{-1}(\cos x)$

$\frac{1}{\cos (x-a) \cos (x-b)}$

$\int \frac{\sin ^2 x-\cos ^2 x}{\sin ^2 x \cos ^2 x} d x$ is equal to
(A) $\tan x+\cot x+\mathrm{C}$
(B) $\tan x+\operatorname{cosec} x+\mathrm{C}$
(C) $-\tan x+\cot x+\mathrm{C}$
(D) $\tan x+\sec x+\mathrm{C}$

$\int \frac{e^x(1+x)}{\cos ^2\left(e^x x\right)} d x$ equals
(A) $-\cot \left(e x^x\right)+\mathrm{C}$
(B) $\tan \left(x e^x\right)+\mathrm{C}$
(C) $\tan \left(e^x\right)+\mathrm{C}$
(D) $\cot \left(e^x\right)+\mathrm{C}$

$\frac{3 x^2}{x^6+1}$

$\frac{1}{\sqrt{1+4 x^2}}$

$\frac{1}{\sqrt{(2-x)^2+1}}$

$\frac{1}{\sqrt{9-25 x^2}}$

$\frac{3 x}{1+2 x^4}$

$\frac{x^2}{1-x^6}$

$\frac{x-1}{\sqrt{x^2-1}}$

$\frac{x^2}{\sqrt{x^6+a^6}}$

$\frac{\sec ^2 x}{\sqrt{\tan ^2 x+4}}$

$\frac{1}{\sqrt{x^2+2 x+2}}$

$\frac{1}{9 x^2+6 x+5}$

$\frac{1}{\sqrt{7-6 x-x^2}}$

$\frac{1}{\sqrt{(x-1)(x-2)}}$

$\frac{1}{\sqrt{8+3 x-x^2}}$

$\frac{1}{\sqrt{(x-a)(x-b)}}$

$\frac{4 x+1}{\sqrt{2 x^2+x-3}}$

$\frac{x+2}{\sqrt{x^2-1}}$

$\frac{5 x-2}{1+2 x+3 x^2}$

$\frac{6 x+7}{\sqrt{(x-5)(x-4)}}$

$\frac{x+2}{\sqrt{4 x-x^2}}$

$\frac{x+2}{\sqrt{x^2+2 x+3}}$

$\frac{x+3}{x^2-2 x-5}$

$\frac{5 x+3}{\sqrt{x^2+4 x+10}}$

$\int \frac{d x}{x^2+2 x+2}$ equals
(A) $x \tan ^{-1}(x+1)+\mathrm{C}$
(B) $\tan ^{-1}(x+1)+\mathrm{C}$
(C) $(x+1) \tan ^{-1} x+\mathrm{C}$
(D) $\tan ^{-1} x+\mathrm{C}$

$\int \frac{d x}{\sqrt{9 x-4 x^2}}$ equals
(A) $\frac{1}{9} \sin ^{-1}\left(\frac{9 x-8}{8}\right)+\mathrm{C}$
(B) $\frac{1}{2} \sin ^{-1}\left(\frac{8 x-9}{9}\right)+\mathrm{C}$
(C) $\frac{1}{3} \sin ^{-1}\left(\frac{9 x-8}{8}\right)+\mathrm{C}$
(D) $\frac{1}{2} \sin ^{-1}\left(\frac{9 x-8}{9}\right)+\mathrm{C}$

$\frac{x}{(x+1)(x+2)}$

$\frac{1}{x^2-9}$

$\frac{3 x-1}{(x-1)(x-2)(x-3)}$

$\frac{x}{(x-1)(x-2)(x-3)}$

$\frac{2 x}{x^2+3 x+2}$

$\frac{1-x^2}{x(1-2 x)}$

$\frac{x}{\left(x^2+1\right)(x-1)}$

$\frac{x}{(x-1)^2(x+2)}$

$\frac{3 x+5}{x^3-x^2-x+1}$

$\frac{2 x-3}{\left(x^2-1\right)(2 x+3)}$

$\frac{5 x}{(x+1)\left(x^2-4\right)}$

$\frac{x^3+x+1}{x^2-1}$

$\frac{2}{(1-x)\left(1+x^2\right)}$

$\frac{3 x-1}{(x+2)^2}$

$\frac{1}{x^4-1}$

$\frac{1}{x\left(x^n+1\right)}$ [Hint: multiply numerator and denominator by $x^{n-1}$ and put $x^n=t$ ]

$\frac{\cos x}{(1-\sin x)(2-\sin x)}$ [Hint : Put $\left.\sin x=t\right]$

$\frac{\left(x^2+1\right)\left(x^2+2\right)}{\left(x^2+3\right)\left(x^2+4\right)}$

$\frac{2 x}{\left(x^2+1\right)\left(x^2+3\right)}$

$\frac{1}{x\left(x^4-1\right)}$

$\frac{1}{\left(e^x-1\right)}\left[\right.$ Hint : Put $\left.e^x=t\right]$

$\int \frac{x d x}{(x-1)(x-2)}$ equals
(A) $\log \left|\frac{(x-1)^2}{x-2}\right|+\mathrm{C}$
(B) $\log \left|\frac{(x-2)^2}{x-1}\right|+\mathrm{C}$
(C) $\log \left|\left(\frac{x-1}{x-2}\right)^2\right|+\mathrm{C}$
(D) $\log |(x-1)(x-2)|+\mathrm{C}$

$\int \frac{d x}{x\left(x^2+1\right)}$ equals
(A) $\log |x|-\frac{1}{2} \log \left(x^2+1\right)+C$
(B) $\log |x|+\frac{1}{2} \log \left(x^2+1\right)+C$
(C) $-\log |x|+\frac{1}{2} \log \left(x^2+1\right)+C$
(D) $\frac{1}{2} \log |x|+\log \left(x^2+1\right)+\mathrm{C}$

$x \sin x$

$x \sin 3 x$

$x^2 e^x$

$x \log x$

$x \log 2 x$

$x^2 \log x$

$x \sin ^{-1} x$

$x \tan ^{-1} x$

$x \cos ^{-1} x$

$\left(\sin ^{-1} x\right)^2$

$\frac{x \cos ^{-1} x}{\sqrt{1-x^2}}$

$x \sec ^2 x$

$\tan ^{-1} x$

$x(\log x)^2$

$\left(x^2+1\right) \log x$

$e^x(\sin x+\cos x)$

$\frac{x e^x}{(1+x)^2}$

$e^x\left(\frac{1+\sin x}{1+\cos x}\right)$

$e^x\left(\frac{1}{x}-\frac{1}{x^2}\right)$

$\frac{(x-3) e^x}{(x-1)^3}$

$e^{2 x} \sin x$

$\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$

$\int x^2 e^{x^3} d x$ equals
(A) $\frac{1}{3} e^{x^3}+\mathrm{C}$
(B) $\frac{1}{3} e^{x^2}+\mathrm{C}$
(C) $\frac{1}{2} e^{x^3}+\mathrm{C}$
(D) $\frac{1}{2} e^{x^2}+\mathrm{C}$

$\int e^x \sec x(1+\tan x) d x$ equals
(A) $e^x \cos x+\mathrm{C}$
(B) $e^x \sec x+\mathrm{C}$
(C) $e^x \sin x+\mathrm{C}$
(D) $e^x \tan x+\mathrm{C}$

$\sqrt{4-x^2}$

$\sqrt{1-4 x^2}$

$\sqrt{x^2+4 x+6}$

$\sqrt{x^2+4 x+1}$

$\sqrt{1-4 x-x^2}$

$\sqrt{x^2+4 x-5}$

$\sqrt{1+3 x-x^2}$

$\sqrt{x^2+3 x}$

$\sqrt{1+\frac{x^2}{9}}$

$\int \sqrt{1+x^2} d x$ is equal to
(A) $\frac{x}{2} \sqrt{1+x^2}+\frac{1}{2} \log \left|\left(x+\sqrt{1+x^2}\right)\right|+\mathrm{C}$
(B) $\frac{2}{3}\left(1+x^2\right)^{\frac{3}{2}}+\mathrm{C}$
(C) $\frac{2}{3} x\left(1+x^2\right)^{\frac{3}{2}}+\mathrm{C}$
(D) $\frac{x^2}{2} \sqrt{1+x^2}+\frac{1}{2} x^2 \log \left|x+\sqrt{1+x^2}\right|+\mathrm{C}$

$\int \sqrt{x^2-8 x+7} d x$ is equal to
(A) $\frac{1}{2}(x-4) \sqrt{x^2-8 x+7}+9 \log \left|x-4+\sqrt{x^2-8 x+7}\right|+\mathrm{C}$
(B) $\frac{1}{2}(x+4) \sqrt{x^2-8 x+7}+9 \log \left|x+4+\sqrt{x^2-8 x+7}\right|+\mathrm{C}$
(C) $\frac{1}{2}(x-4) \sqrt{x^2-8 x+7}-3 \sqrt{2} \log \left|x-4+\sqrt{x^2-8 x+7}\right|+\mathrm{C}$
(D) $\frac{1}{2}(x-4) \sqrt{x^2-8 x+7}-\frac{9}{2} \log \left|x-4+\sqrt{x^2-8 x+7}\right|+\mathrm{C}$

$x \sqrt{x+x^2}$

$x+1 \sqrt{2 x^2+3}$

$(x+3) \sqrt{3-4 x-x^2}$

$\int_{-1}^1(x+1) d x$

$\int_2^3 \frac{1}{x} d x$

$\int_1^2\left(4 x^3-5 x^2+6 x+9\right) d x$

$\int_0^{\frac{\pi}{4}} \sin 2 x d x$

$\int_0^{\frac{\pi}{2}} \cos 2 x d x$

$\int_4^5 e^x d x$

$\int_0^{\frac{\pi}{4}} \tan x d x$

$\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \operatorname{cosec} x d x$

$\int_0^1 \frac{d x}{\sqrt{1-x^2}}$

$\int_0^1 \frac{d x}{1+x^2}$

$\int_2^3 \frac{d x}{x^2-1}$

$\int_0^{\frac{\pi}{2}} \cos ^2 x d x$

$\int_2^3 \frac{x d x}{x^2+1}$

$\int_0^1 \frac{2 x+3}{5 x^2+1} d x$

$\int_0^1 x e^{x^2} d x$

$\int_1^2 \frac{5 x^2}{x^2+4 x+3}$

$\int_0^{\frac{\pi}{4}}\left(2 \sec ^2 x+x^3+2\right) d x$

$\int_0^\pi\left(\sin ^2 \frac{x}{2}-\cos ^2 \frac{x}{2}\right) d x$

$\int_0^2 \frac{6 x+3}{x^2+4} d x$

$\int_0^1\left(x e^x+\sin \frac{\pi x}{4}\right) d x$

$\int_1^{\sqrt{3}} \frac{d x}{1+x^2}$ equals
(A) $\frac{\pi}{3}$
(B) $\frac{2 \pi}{3}$
(C) $\frac{\pi}{6}$
(D) $\frac{\pi}{12}$

$\int_0^{\frac{2}{3}} \frac{d x}{4+9 x^2}$ equals
(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{12}$
(C) $\frac{\pi}{24}$
(D) $\frac{\pi}{4}$

$\int_0^1 \frac{x}{x^2+1} d x$

$\int_0^{\frac{\pi}{2}} \sqrt{\sin \phi} \cos ^5 \phi d \phi$

$\int_0^1 \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) d x$

$\int_0^2 x \sqrt{x+2}$ (Put $x+2=t^2$ )

$\int_0^{\frac{\pi}{2}} \frac{\sin x}{1+\cos ^2 x} d x$

$\int_0^2 \frac{d x}{x+4-x^2}$

$\int_{-1}^1 \frac{d x}{x^2+2 x+5}$

$\int_1^2\left(\frac{1}{x}-\frac{1}{2 x^2}\right) e^{2 x} d x$

The value of the integral $\int_{\frac{1}{3}}^1 \frac{\left(x-x^3\right)^{\frac{1}{3}}}{x^4} d x$ is
(A) 6
(B) 0
(C) 3
(D) 4

If $f(x)=\int_0^x t \sin t d t$, then $f^{\prime}(x)$ is
(A) $\cos x+x \sin x$
(B) $x \sin x$
(C) $x \cos x$
(D) $\sin x+x \cos x$

$\int_0^{\frac{\pi}{2}} \cos ^2 x d x$

$\int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$

$\int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x d x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}$

$\int_0^{\frac{\pi}{2}} \frac{\cos ^5 x d x}{\sin ^5 x+\cos ^5 x}$

$\int_{-5}^5|x+2| d x$

$\int_2^8|x-5| d x$

$\int_0^1 x(1-x)^n d x$

$\int_0^{\frac{\pi}{4}} \log (1+\tan x) d x$

$\int_0^2 x \sqrt{2-x} d x$

$\int_0^{\frac{\pi}{2}}(2 \log \sin x-\log \sin 2 x) d x$

$\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sin ^2 x d x$

$\int_0^\pi \frac{x d x}{1+\sin x}$

$\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sin ^7 x d x$

$\int_0^{2 \pi} \cos ^5 x d x$

$\int_0^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1+\sin x \cos x} d x$

$\int_0^\pi \log (1+\cos x) d x$

$\int_0^a \frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}} d x$

$\int_0^4|x-1| d x$

Show that $\int_0^a f(x) g(x) d x=2 \int_0^a f(x) d x$, if $f$ and $g$ are defined as $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$

The value of $\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\left(x^3+x \cos x+\tan ^5 x+1\right) d x$ is
(A) 0
(B) 2
(C) $\pi$
(D) 1

The value of $\int_0^{\frac{\pi}{2}} \log \left(\frac{4+3 \sin x}{4+3 \cos x}\right) d x$ is
(A) 2
(B) $\frac{3}{4}$
(C) 0
(D) -2

Example 1
Write an anti derivative for each of the following functions using
the method of inspection:
(i) cos 2x

Example 1
Write an anti derivative for each of the following functions using the method of inspection:
(ii) $3 x^2+4 x^3$

Example 1
Write an anti derivative for each of the following functions using the method of inspection:
(iii) $\frac{1}{x}, x \neq 0$

Example 2
Find the following integrals:
(i) $\int \frac{x^3-1}{x^2} d x$

Example 2
Find the following integrals:
(ii) $\int\left(x^{\frac{2}{3}}+1\right) d x$

Example 2
Find the following integrals:
(iii) $\int\left(x^{\frac{3}{2}}+2 e^x-\frac{1}{x}\right) d x$

Example 3
Find the following integrals:
(i) $\int(\sin x+\cos x) d x$

Example 3
Find the following integrals:
(ii) $\int \operatorname{cosec} x(\operatorname{cosec} x+\cot x) d x$

Example 3
Find the following integrals:
(iii) $\int \frac{1-\sin x}{\cos ^2 x} d x$

Example 4
Find the anti derivative F of $f$ defined by $f(x)=4 x^3-6$, where $\mathrm{F}(0)=3$

Example 5
Integrate the following functions w.r. t. x:
(i) $\sin m x$

Example 5
Integrate the following functions w.r. t. x:
(ii) $2 x \sin \left(x^2+1\right)$

Example 5
Integrate the following functions w.r. t. x:
(iii) $\frac{\tan ^4 \sqrt{x} \sec ^2 \sqrt{x}}{\sqrt{x}}$

Example 5
Integrate the following functions w.r. t. x:
(iv) $\frac{\sin \left(\tan ^{-1} x\right)}{1+x^2}$

Example 6
Find the following integrals:
(i) $\int \sin ^3 x \cos ^2 x d x$

Example 6
Find the following integrals:
(ii) $\int \frac{\sin x}{\sin (x+a)} d x$

Example 6
Find the following integrals:
(iii) $\int \frac{1}{1+\tan x} d x$

Example 7
Find
(i) $\int \cos ^2 x d x$

Example 7
Find
(ii) $\int \sin 2 x \cos 3 x d x$

Example 7
Find
(iii) $\int \sin ^3 x d x$

Example 8
Find the following integrals:
(i) $\int \frac{d x}{x^2-16}$

Example 8
Find the following integrals:
(ii) $\int \frac{d x}{\sqrt{2 x-x^2}}$

Example 9
Find the following integrals:
(i) $\int \frac{d x}{x^2-6 x+13}$

Example 9
Find the following integrals:
(ii) $\int \frac{d x}{3 x^2+13 x-10}$

Example 9
Find the following integrals:
(iii) $\int \frac{d x}{\sqrt{5 x^2-2 x}}$

Example 10
Find the following integrals:
(i) $\int \frac{x+2}{2 x^2+6 x+5} d x$

Example 10
Find the following integrals:
(ii) $\int \frac{x+3}{\sqrt{5-4 x-x^2}} d x$

Example 11
Find $\int \frac{d x}{(x+1)(x+2)}$

Example 12
Find $\int \frac{x^2+1}{x^2-5 x+6} d x$

Example 13
Find $\int \frac{3 x-2}{(x+1)^2(x+3)} d x$

Example 14
Find $\int \frac{x^2}{\left(x^2+1\right)\left(x^2+4\right)} d x$

Example 15
Find $\int \frac{(3 \sin \phi-2) \cos \phi}{5-\cos ^2 \phi-4 \sin \phi} d \phi$

Example 16
Find

$$
\int \frac{x^2+x+1}{(x+2)(x^2+1)} \, dx.
$$

Example 17
Find

$$
\int x\cos x \, dx.
$$

Example 18
Find

$$
\int \log x \, dx.
$$

Example 19
Find

$$
\int xe^x \, dx.
$$

Example 20
Find

$$
\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}} \, dx.
$$

Example 21
Find

$$
\int e^x\sin x \, dx.
$$

Example 22
Find:
(i)
$$
\int e^x\left(\tan^{-1}x+\frac{1}{1+x^2}\right) \, dx.
$$

(ii)

$$
\int \frac{(x^2+1)e^x}{(x+1)^2} \, dx.
$$

Example 23
Find

$$
\int \sqrt{x^2+2x+5} \, dx.
$$

Example 24
Find

$$
\int \sqrt{3-2x-x^2} \, dx.
$$

Example 25
Evaluate the following integrals:
(i)
$$
\int_2^3 x^2 \, dx.
$$

Example 25
Evaluate the following integrals:
(ii) $\int_4^9 \frac{\sqrt{x}}{\left(30-x^{\frac{3}{2}}\right)^2} d x$

Example 25
Evaluate the following integrals:
(iii)
$$
\int_1^2 \frac{xdx}{(x+1)(x+2)} \
$$

Example 25
Evaluate the following integrals:
(iv)
$$
\int_0^{\frac{\pi}{4}} \sin^3 2t \cos 2t \, dt.
$$

Example 26 (Method 1)
Evaluate
$$
\int_{-1}^{1} 5x^4\sqrt{x^5+1} \, dx.
$$

Example 27 (Method 1)
Evaluate
$$
\int_0^1 \frac{\tan^{-1}x}{1+x^2} \, dx.
$$

Example 28
Evaluate
$$
\int_{-1}^{2} \left|x^3-x\right| \, dx.
$$

Example 29
Evaluate
$$
\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin^2 x \, dx.
$$

Example 30
Evaluate
$$
\int_0^\pi \frac{x\sin x}{1+\cos^2 x} \, dx.
$$

Example 31
Evaluate
$$
\int_{-1}^{1} \sin^5 x \cos^4 x \, dx.
$$

Example 32
Evaluate
$$
\int_0^{\frac{\pi}{2}} \frac{\sin^4 x}{\sin^4 x+\cos^4 x} \, dx.
$$

Example 33 (Method 1)
Evaluate
$$
\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1+\sqrt{\tan x}}.
$$

Example 34
Evaluate
$$
\int_0^{\frac{\pi}{2}} \log(\sin x) \, dx.
$$

Example 35
Find
$$
\int \cos 6x \sqrt{1+\sin 6x} \, dx.
$$

Example 36
Find
$$
\int \frac{(x^4-x)^{\frac{1}{4}}}{x^5} \, dx.
$$

Example 37
Find
$$
\int \frac{x^4}{(x-1)(x^2+1)} \, dx.
$$

Example 38
Find
$$
\int \left[\log(\log x)+\frac{1}{(\log x)^2}\right] \, dx.
$$

Example 39
Find
$$
\int \left[\sqrt{\cot x}+\sqrt{\tan x}\right] \, dx.
$$

Example 40
Find
$$
\int \frac{\sin 2x \cos 2x}{\sqrt{9-\cos^4(2x)}} \, dx.
$$

Example 41 (Introduction)
Evaluate
$$
\int_{-1}^{\frac{3}{2}} \left|x\sin(\pi x)\right| \, dx.
$$

Example 42
Evaluate
$$
\int_0^\pi \frac{x}{a^2\cos^2 x+b^2\sin^2 x} \, dx.
$$

Misc 1 (Method 1)
$\frac{1}{x-x^3}$

Misc 2
$\frac{1}{\sqrt{x+a}+\sqrt{x+b}}$

Misc 3
$\frac{1}{x \sqrt{a x-x^2}}$ [Hint:Put $x=\frac{a}{t}$ ]

Misc 4
$\frac{1}{x^2\left(x^4+1\right)^{\frac{3}{4}}}$

Misc 5
$\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}$ [Hint: $\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}=\frac{1}{x^{\frac{1}{3}}\left(1+x^{\frac{1}{6}}\right)}$, put $x=t^6$ ]

Misc 6
$\frac{5 x}{(x+1)\left(x^2+9\right)}$

Misc 7
$\frac{\sin x}{\sin (x-a)}$

Misc 8
$\frac{e^{5 \log x}-e^{4 \log x}}{e^{3 \log x}-e^{2 \log x}}$

Misc 9
$\frac{\cos x}{\sqrt{4-\sin ^2 x}}$

Misc 10
$\frac{\sin ^8-\cos ^8 x}{1-2 \sin ^2 x \cos ^2 x}$

Misc 11
$\frac{1}{\cos (x+a) \cos (x+b)}$

Mise 12
$\frac{x^3}{\sqrt{1-x^8}}$

Misc 13
$\frac{e^x}{\left(1+e^x\right)\left(2+e^x\right)}$

Misc 14
$\frac{1}{\left(x^2+1\right)\left(x^2+4\right)}$

Misc 15
$\cos ^3 x e^{\log \sin x}$

Misc 16
$e^{3 \log x}\left(x^4+1\right)^{-1}$

Misc 17
$f^{\prime}(a x+b)[f(a x+b)]^n$

Misc 18
$\frac{1}{\sqrt{\sin ^3 x \sin (x+\alpha)}}$

Misc 19
$\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$

Misc 20
$\frac{2+\sin 2 x}{1+\cos 2 x} e^x$

Misc 21
$\frac{x^2+x+1}{(x+1)^2(x+2)}$

Misc 22
$\tan ^{-1} \sqrt{\frac{1-x}{1+x}}$

Misc 23
$\frac{\sqrt{x^2+1}\left[\log \left(x^2+1\right)-2 \log x\right]}{x^4}$

Misc 24
$\int_{\frac{\pi}{2}}^\pi e^x\left(\frac{1-\sin x}{1-\cos x}\right) d x$

Misc 25
$\int_0^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos ^4 x+\sin ^4 x}$

Misc 26
$\int_0^{\frac{\pi}{2}} \frac{\cos ^2 x d x}{\cos ^2 x+4 \sin ^2 x}$

Misc 27
$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x+\cos x}{\sqrt{\sin 2 x}} d x$

Misc 28
$\int_0^1 \frac{d x}{\sqrt{1+x}-\sqrt{x}}$

Misc 29
$\int_0^{\frac{\pi}{4}} \frac{\sin x+\cos x}{9+16 \sin 2 x} d x$

Misc 30
$\int_0^{\frac{\pi}{2}} \sin 2 x \tan ^{-1}(\sin x) d x$

Misc 31
$\int_1^4[|x-1|+|x-2|+|x-3|] d x$

Misc 32
$\int_1^3 \frac{d x}{x^2(x+1)}=\frac{2}{3}+\log \frac{2}{3}$

Misc 33
$\int_0^1 x e^x d x=1$

Misc 34
$\int_{-1}^1 x^{17} \cos ^4 x d x=0$

Misc 35
$\int_0^{\frac{\pi}{2}} \sin ^3 x d x=\frac{2}{3}$

Misc 36
$\int_0^{\frac{\pi}{4}} 2 \tan ^3 x d x=1-\log 2$

Misc 37
$\int_0^1 \sin ^{-1} x d x=\frac{\pi}{2}-1$

Misc 38
$\int \frac{d x}{e^x+e^{-x}}$ is equal to
(A) $\tan ^{-1}\left(e^x\right)+\mathrm{C}$
(B) $\tan ^{-1}\left(e^{-x}\right)+\mathrm{C}$
(C) $\log \left(e^x-e^{-x}\right)+\mathrm{C}$
(D) $\log \left(e^x+e^{-x}\right)+\mathrm{C}$

Misc 39
$\int \frac{\cos 2 x}{(\sin x+\cos x)^2} d x$ is equal to
(A) $\frac{-1}{\sin x+\cos x}+\mathrm{C}$
(B) $\log |\sin x+\cos x|+\mathrm{C}$
(C) $\log |\sin x-\cos x|+\mathrm{C}$
(D) $\frac{1}{(\sin x+\cos x)^2}$

Misc 40
If $f(a+b-x)=f(x)$, then $\int_a^b x f(x) d x$ is equal to
(A) $\frac{a+b}{2} \int_a^b f(b-x) d x$
(B) $\frac{a+b}{2} \int_a^b f(b+x) d x$
(C) $\frac{b-a}{2} \int_a^b f(x) d x$
(D) $\frac{a+b}{2} \int_a^b f(x) d x$