Question:
Write a value of $\int \sqrt{4-\mathrm{x}^{2}} \mathrm{dx}$.
Solution:
we know that $\int \sqrt{a^{2}-x^{2}} d x=\frac{x \sqrt{a^{2}-x^{2}}}{2}+\frac{x^{2}}{2} \sin ^{-1}\left(\frac{x}{a}\right)+c$
Given $\int \sqrt{4-x^{2}}$
$=\int \sqrt{2^{2}-x^{2}}$
$=\frac{x \sqrt{2^{2}-x^{2}}}{2}+\frac{x^{2}}{2} \sin ^{-1}\left(\frac{x}{2}\right)$
$=\frac{x \sqrt{4-x^{2}}}{2}+\frac{x^{2}}{2} \sin ^{-1}\left(\frac{x}{2}\right)+c$