Question

Evaluate the integral:

$\int \sqrt{a^{2} -x^{2}} dx$

Answer 1

$\frac{x}{2}\sqrt{a^{2} -x^{2}} + \frac{a^{2}}{2}\sin^{-1}\left(\frac{x}{a}\right) + C$

Answer 2

The integral $\int \sqrt{a^{2} -x^{2}} dx$ is a standard form. The result is obtained by applying the known formula or by using the trigonometric substitution $x = a \sin \theta$, evaluating the resulting integral in terms of $\theta$, and then substituting back to express the result in terms of $x$. The constant of integration $C$ is added as it is an indefinite integral.