Calculus Examples

$\int \sqrt{9 - x^{2}} d x$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 - x^{2}}$ as ${(9 - x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\int {(9 - x^{2})}^{\frac{1}{2}} d x]$

Step 2

$\int {(9 - x^{2})}^{\frac{1}{2}} d x$ is an antiderivative of ${(9 - x^{2})}^{\frac{1}{2}}$ , so by definition $\frac{d}{d x} [\int {(9 - x^{2})}^{\frac{1}{2}} d x]$ is ${(9 - x^{2})}^{\frac{1}{2}}$ .

${(9 - x^{2})}^{\frac{1}{2}}$

Step 3

Reorder terms.

${(- x^{2} + 9)}^{\frac{1}{2}}$