Calculus Examples

$\int x^{2} \sqrt{1 + x} d x$

Step 1

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

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

Step 2

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

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