Evaluate the following integrals:


Evaluate $\int \sqrt{a^{2}-x^{2}} d x$


Let, $x=a \sin t$

Differentiate both side with respect to $t$

$\frac{d x}{d t}=a \cos t \Rightarrow \mathrm{dx}=\mathrm{a} \cos \mathrm{t} \mathrm{dt}$

$y=\int \sqrt{a^{2}-(a \sin t)^{2}} a \cos t d t$

$y=\int(a \cos t)(a \cos t) d t$

$y=\int a^{2} \cos ^{2} t d t$

$y=\int a^{2}\left(\frac{1+\cos 2 t}{2}\right) d t$

$y=\frac{a^{2}}{2} \int 1+\cos 2 t d t$

$y=\frac{a^{2}}{2}\left(t+\frac{\sin 2 t}{2}\right)+c$

Again, put $t=\sin ^{-1} \frac{x}{a}$

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

$y=\frac{a^{2}}{2}\left(\sin ^{-1} \frac{x}{a}+\frac{2 \times \frac{x}{a} \times \sqrt{1-\frac{x^{2}}{a^{2}}}}{2}\right)+c$

$y=\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}+\frac{x}{2} \sqrt{a^{2}-x^{2}}+c$

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