Exercise 19.7
Question 1
\int \frac{\cos \sqrt{x}}{\sqrt{x}} d xSol :
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
\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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