Question

Find the integral of following examples. Only correct answer No Spam. I will mark as BRAINLIST.

Answer 1

12. Given to find,

$\displaystyle\longrightarrow I=\int\dfrac{\pi}{\sqrt{\pi-x^2}}\ dx$

Taking the constant $\pi$ out of the integral,

$\displaystyle\longrightarrow I=\pi\int\dfrac{1}{\sqrt{\pi-x^2}}\ dx\quad\quad\dots(1)$

Substitute,

$\longrightarrow x=\sqrt\pi\sin\theta$

$\longrightarrow dx=\sqrt\pi\cos\theta\ d\theta$

Then (1) becomes,

$\displaystyle\longrightarrow I=\pi\int\dfrac{\sqrt\pi\cos\theta}{\sqrt{\pi-\pi\sin^2\theta}}\ d\theta$

$\displaystyle\longrightarrow I=\pi\int\dfrac{\cos\theta}{\sqrt{1-\sin^2\theta}}\ d\theta$

$\displaystyle\longrightarrow I=\pi\int1\ d\theta\quad[\,\because\sin^2\theta+\cos^2\theta=1\,]$

$\displaystyle\longrightarrow I=\pi\theta+C$

$\displaystyle\longrightarrow\underline{\underline{I=\pi\sin^{-1}\left(\dfrac{x}{\sqrt\pi}\right)+C}}$

13. Given to find,

$\displaystyle\longrightarrow I=\int(2\cos x-5\sin x)\ dx$

Applying linearity,

$\displaystyle\longrightarrow I=\int2\cos x\ dx-\int5\sin x\ dx$

Taking constants out of the integral,

$\displaystyle\longrightarrow I=2\int\cos x\ dx-5\int\sin x\ dx$

$\displaystyle\longrightarrow\underline{\underline{I=2\sin x+5\cos x+C}}$

14. Given to find,

$\displaystyle\longrightarrow I=\int\dfrac{1}{\sqrt{1-\dfrac{x^2}{2}}}\ dx$

Or,

$\displaystyle\longrightarrow I=\int\dfrac{1}{\sqrt{\dfrac{2-x^2}{2}}}\ dx$

$\displaystyle\longrightarrow I=\int\dfrac{\sqrt2}{\sqrt{2-x^2}}\ dx$

Taking the constant out of the integral,

$\displaystyle\longrightarrow I=\sqrt2\int\dfrac{1}{\sqrt{2-x^2}}\ dx\quad\quad\dots(2)$

Substitute,

$\longrightarrow x=\sqrt2\,\sin\theta$

$\longrightarrow dx=\sqrt2\,\cos\theta\ d\theta$

Then (2) becomes,

$\displaystyle\longrightarrow I=\sqrt2\int\dfrac{\sqrt2\,\cos\theta}{\sqrt{2-2\sin^2\theta}}\ d\theta$

$\displaystyle\longrightarrow I=\sqrt2\int\dfrac{\cos\theta}{\sqrt{1-\sin^2\theta}}\ d\theta$

$\displaystyle\longrightarrow I=\sqrt2\int1\ d\theta\quad[\,\because\sin^2\theta+\cos^2\theta=1\,]$

$\displaystyle\longrightarrow I=\theta\sqrt2+C$

$\displaystyle\longrightarrow\underline{\underline{I=\sqrt2\,\sin^{-1}\left(\dfrac{x}{\sqrt2}\right)+C}}}}$