Question

Integrate the function $\sqrt{4-x^2}$

Answer 1

Let $I=\int \sqrt{4-x^2}dx=\int \sqrt{(2{)}^2-(x{)}^2}dx$
We know that, $\int \sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}{\sin }^{-1}\frac{x}{a}+C$
$\therefore I=\frac{x}{2}\sqrt{4-x^2}+\frac{4}{2}{\sin }^{-1}\frac{x}{2}+C$
$=\frac{x}{2}\sqrt{4-x^2}+2{\sin }^{-1}\frac{x}{2}+C$