KC Sinha Solution Class 12 Chapter 19 Indefinite Integrals Exercise 19.7

Exercise 19.7

Question 1
$\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x$
Sol :
Let z=√x then $d z=\frac{1}{2 \sqrt{x}} d x$
$2 d z=\frac{1}{\sqrt{x}} d x$

Now , $\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x$

$=\int \cos z \cdot 2 d z$

$=2 \int \cos z d z$

=2sinz+c

=2sin√x+c

Question 2
$\int \frac{1}{x^{2}} \cdot \sin \frac{1}{x} d x$
Sol :
Let $z=\frac{1}{x}$ then $d z=-\frac{1}{x^{2}} d x$
$\Rightarrow-d z=\frac{1}{x^{2}} d x$

Now , $\int \frac{1}{x^{2}} \sin \frac{1}{x} d x$

$=\int \sin \frac{1}{x} \cdot \frac{1}{x^{2}} d x$

$=\int \sin z(-d z)$

$=-\int \sin zd z$

=-(-cos z)+c

=cosz+c

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


Question 3
$\int \frac{1}{x^{2}} \cos \frac{1}{x} d x$
Sol :
Let $z=\frac{1}{x}$ then $d z=-\frac{1}{x^{2}} d x$

$\Rightarrow-d z=\frac{1}{x^{2}} d x$

Now , $\int \frac{1}{x^{2}} \cos \frac{1}{x} d x$

$=\int \cos \frac{1}{x} \cdot \frac{1}{x^{2}} d x$

$=\int \cos z(-d z)$

$=-\int \cos z d z$

=-sinz+c

$=-\sin \frac{1}{x}+c$

Question 4
$\int e^{x} \cdot \cos \left(e^{x}+2\right) d x$
Sol :
Let $z=e^{x}+2$ then $d z=e^{x} d x$

Now , $\int e^{x} \cdot \cos \left(e^{x}+2\right) d x$

$=\int \cos \left(e^{x}+2\right) \cdot e^{x} d x$

$=\int \cos z \cdot d z$

=sinz+c

$=\sin \left(e^{x}+2\right)+c$


Question 5
$\int \frac{\sin \sqrt{x+1}}{\sqrt{x+1}} d x$
Sol :
Let $z=\sqrt{x+1}$ then $d z=\frac{1}{2 \sqrt{x+1}} d x$

$2 d z=\frac{1}{\sqrt{x+1}} d x$

Now , $\int \frac{\sin \sqrt{x+1}}{\sqrt{x+1}} d x$

$=\int \sin z \cdot z d z$

$=2 \int \sin z d z$

=2(-cosz)+c

=-2cosz+c

=$-2 \cos \sqrt{x+1}+c$


Question 6
$\int x^{2} \cdot \sec x^{3} d x$
Sol :
Let $z=x^{3}$ then $d z=3 x^{2} d x$

$\frac{d z}{3}=x^{2} d x$

Now , $\int x^{2} \cdot \sec x^{3} d x$

$=\int \sec x^{3} \cdot x^{2} d x$

$=\int \sec z \cdot \frac{d z}{3}$

$=\frac{1}{3} \int \sec z d z$

$=\frac{1}{3} \log |\sec z+\tan z|+c$

$=\frac{1}{3} \log \left|\sec x^{3}+\tan x^{3}\right|+c$


Question 7
$\int x^{\frac{1}{3}} \cdot \sin x^{\frac{4}{3}} d x$
Sol :
let $z=x^{\frac{4}{3}}$ then $d z=\frac{4}{3} \cdot x^{\frac{1}{3}} d x$
$\frac{3}{4} d z=x^{\frac{1}{3}} d x$

Now , $\int x^{\frac{1}{3}} \cdot \sin x^{\frac{4}{3}} d x$

$=\int \sin x^{\frac{4}{3}} x^{\frac{1}{3}} d x$

$=\int \sin z \cdot \frac{3}{4} d z$

$=\frac{3}{4} \int \sin z d z$

$=\frac{3}{4}(-\cos z)+c$

$=\frac{-3}{4} \cos z+c$

$=-\frac{3}{4} \cos x^{\frac{4}{3}}+c$



Question 8
$\int\left(x^{2}+1\right) \cdot \cos \left(x^{3}+3 x+2\right) d x$
Sol :
Let $z=x^{3}+3 x+2$ then $d z=\left(3 x^{2}+3\right) d x=3\left(x^{2}+1\right) d x$
$d z=3\left(x^{2}+1\right) d x$
$\frac{d z}{3}=\left(x^{2}+1\right) d x$

Now , $\int\left(x^{2}+1\right) \cdot \cos \left(x^{3}+3 x+2\right) d x$

$=\int \cos \left(x^{3}+3 x+2\right) \cdot\left(x^{2}+1\right) d x$

$=\int \cos z \frac{d z}{3}$

$=\frac{1}{3} \int \cos z d 2$

$=\frac{1}{3} \sin z+c$

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


Question 9
$\int \frac{\cos (\log_e x)}{x} d x$
Sol :
Let z=logx then $d z=\frac{1}{x} d x$

Now , $\int \frac{\cos (\log x)}{x} d x$

$=\int \cos z\cdot d z$

=sinz+c

=sin(logx)+c

Question 10
(i) $\int \frac{\sec ^{2}(\log x)}{x} d x$
Sol :
Let z=logx then $d z=\frac{1}{x} d x$

Now , $\int \frac{\sec ^{2}(\log x)}{x} d x$

$=\int \sec ^{2} z d z$

=tanz+c

=tan(logx)+c


(ii) $\int \frac{\operatorname{cosec}^{2}(\log x)}{x} d x$
Sol :
Let z=logx then $d z=\frac{1}{x} d x$

Now  , $\int \frac{\operatorname{cosec}^{2}(\log x)}{x} d x$

$=\int \operatorname{cosec}^{2} z d z$

=-cotz+c

=-cot(logx)+c

Question 11
$\int \frac{\sin (2+3 \log x)}{x} d x$
Sol :
Let z=2+3logx then $d z=3 \frac{1}{x} d x$
$d z=\frac{3}{x} d x$ $\Rightarrow \frac{d z}{3}=\frac{1}{x} d x$

Now , $\int \frac{\sin (2+3 \log x)}{x} d x$ $=\int \sin z \cdot \frac{d z}{3}$

$=\frac{1}{3} \int \sin z d z$ $=\frac{1}{3}(-\cos z)+c$ $=\frac{-1}{3} \cos (2+3 \log x)+c$

Question 12
$\int \frac{\tan \sqrt{x} \cdot \sec ^{2} \sqrt{x}}{\sqrt{x}} d x$
Sol :
Let $z=\tan \sqrt{x}$ then $d z=\frac{\sec ^{2} \sqrt{x}}{2 \sqrt{x}} d x$
$2 d z=\frac{\sec ^{2} \sqrt{x}}{\sqrt{x}} d x$

Now, $\int \frac{\tan \sqrt{x}-\sec ^{2} \sqrt{x}}{\sqrt{x}} d x$

$=\int z\cdot 2 d z$

$=2 \int z d z$

$=2 \frac{z^{2}}{2}+c$
$=z^{2}+c$
$=\tan ^{2}\sqrt{x}+c$


Question 13
$\int \frac{1}{x \cos ^{2}(\log x)} d x$
Sol :
Let z=logx then $d z=\frac{1}{x} d x$

Now , $\int \frac{1}{x \cos ^{2}(\log x)} d x$

$=\int \frac{\sec ^{2}(\log x)}{x} d x$

$=\int \sec ^{2} z d z$
=tanz+c
=tan(logx)+c



Question 14
$\int e^{x} \cdot \tan e^{x} \cdot \sec e^{x} d x$
Sol :
Let $z=e^{x}$ then $d z=e^{x} d x$

Now , $\int e^{x} \cdot \tan e^{x} \cdot \sec e^{x} d x$

$=\int \sec e^{x} \cdot \tan e^{x} \cdot e^{x} d x$

$=\int \sec z \cdot \tan z d z$
=secz+c
$=\sec e^{x}+c$



Question 15
$\int \frac{\sec ^{2} \sqrt{x+1}}{\sqrt{x+1}} d x$
Sol :
Let $z=\sqrt{x+1}$ then $d z=\frac{1}{2 \sqrt{x+1}} d x$
$2 d z=\frac{1}{\sqrt{x+1}} d x$

Now , $\int \frac{\sec ^{2} \sqrt{x+1}}{\sqrt{x+1}} d x$

$=\int \sec ^{2} z \cdot 2 d z$

$=2 \int \sec ^{2} z d z$

=2tanz+c
$=2 \tan \sqrt{x+1}+c$


Question 16
$\int 2 x \cdot \sin \left(x^{2}+1\right) d x$
Sol :
Let $z=x^{2}+1$ then dz=2xdx

Now , $\int 2 x \cdot \sin \left(x^{2}+1\right) d x$

$=\int \sin \left(x^{2}+1\right) \cdot 2 x d x$

$=\int \sin z \cdot d z$
$=(-\cos z)+c$
$=-\cos \left(x^{2}+1\right)+c$

Question 17
$\int \sin x \cdot \sin (\cos x) d x$
Sol :
Let z=cosx then dz=-sinxdx
⇒-dz=sinxdx

Now , $\int \sin x \cdot \sin (\cos x) d x$

$=\int \sin (\cos x) \cdot \sin x d x$

$=\int \sin z(-d z)$

$=-\int \sin z d z$

$=-(-\cos z)+c$

=cosz+c

=cos(cosx)+c

Question 18
$\int \frac{e^{x}(1+x)}{\sin ^{2}\left(x e^{x}\right)} d x$
Sol :
Let $z=x e^{x}$  then $d z=\left(x e^{x}+e^{x}\right) d x$
$d z=e^{x}(1+x) d x$

Now , $\int \frac{e^{x}(1+x) d x}{\sin ^{2}\left(x e^{x}\right)}$

$=\int \frac{d z}{\sin ^{2} z}$

$=\int \operatorname{cosec}^{2} z d z$

=-cotz+c
$=-\cot \left(x e^{x}\right)+c$


Question 19
$\int \frac{\cos \sqrt{x}-3}{\sqrt{x}} d x$
Sol :
Let $z=\sqrt{x}-3$ then $d z=\frac{1}{2 \sqrt{x}} d x$
$2 d z=\frac{1}{\sqrt{x}} d x$

Now , $\int \frac{\cos \sqrt{x}-3}{\sqrt{x}} d x$

$=\int \cos z \cdot 2 d z=$

$=2 \int \cos z d z$

=2sinz+c

=$=2 \sin (\sqrt{x}-3)+c$


Question 20
$\int \frac{\sec ^{2}(2+\log x)}{x} d x$
Sol :
Let z=(2+logx) then d z=\frac{1}{x} d x

Now , $\int \frac{\sec ^{2}(2+\log x)}{x} d x$

$=\int \sec ^{2} z d z$

=tanz+c
=tan(2+logx)+c

Question 21
$\int \frac{\cos \sqrt{a x+b}}{\sqrt{a x+b}} d x$
Sol :
Let $z=\sqrt{a x+b}$ then $d z=\frac{1 \times a}{2 \sqrt{a x+b}} d x$
$\Rightarrow \frac{2}{a} d z=\frac{1}{\sqrt{a x+b}} d x$

Now , $\int \frac{\cos \sqrt{a x}+b}{\sqrt{a x+b}} d x$

$=\int \cos z \cdot \frac{2}{a} d z$

$=\frac{2}{a} \int \cos z d z$

$=\frac{2}{a} \sin z+c$

$=\frac{2}{a} \sin \sqrt{a x+b}+c$


Question 22
$\int \frac{\sin ^{3}(3+2 \log x)}{x} d x$
Sol :
Let z=(3+2logx) then $d 2=\frac{2}{x} d x \quad \Rightarrow \frac{d z}{2}=\frac{1}{x} d x$

Now , $\int \frac{\sin ^{3}(3+2 \log x)}{x} d x$

$=\int \sin ^{3} z \cdot \frac{d z}{2}$

$=\frac{1}{2} \int \sin ^{3} z d 2$

$=\frac{1}{2} \int \frac{3 \sin z-\sin 3z}{4} d z$

$=\frac{3}{8} \int \sin z d z-\frac{1}{8} \int \sin 3z d z$

$=-\frac{3}{8} \cos z+\frac{1}{8 \times 3} \cos 3z+c$

$=\frac{-3}{8} \cos z+\frac{1}{24} \cos 3z+c$

$-\frac{3}{8} \cos (3+2 \log x)+\frac{1}{24} \cos 3(3+2 \log x)+c$

$=-\frac{3}{8} \cos (3+2 \log x)+\frac{1}{24} \cos (9+6 \log x)+c$


Question 23
$\int \frac{\sin \left(\tan ^{-1} x\right)}{1+x^{2}} d x$
Sol :
Let $z=\tan ^{2} x$ then $d z=\frac{1}{1+x^{2}} d x$

Now , $\int \frac{\sin \left(\tan ^{-1} x\right)}{1+x^{2}} d x$

$=\int \sin z d z$

=-cosz+c
$=-\cos \left(\tan ^{-1} x\right)+c$


Question 24
$\int \frac{x^{3} \cdot \sin \left(\tan ^{-1} x^{4}\right)}{1+x^{8}} d x$
Sol :
Let $z=\tan ^{-1} x^{4}$ then

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

$d z=\frac{4 x^{3}}{1+x^{8}} d x$

$\frac{d z}{4}=\frac{x^{3}}{1+x^{8}} d x$

Now  , $\int \frac{x^{3} \cdot \sin \left(\tan ^{-1} x^{4}\right)}{1+x^{8}} d x$

$=\int \sin z \cdot \frac{d z}{4}=\frac{1}{4} \int \sin z d z$

$=\frac{1}{4}(-\cos z)+c$

$=-\frac{1}{4} \cos \left(\tan ^{-1} x^{4}\right)+c$


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