Calculus Examples

$\int 2 x \sqrt{7 + x^{2}} d x$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 + x^{2}}$ as ${(7 + x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\int 2 x {(7 + x^{2})}^{\frac{1}{2}} d x]$

Step 2

$\int 2 x {(7 + x^{2})}^{\frac{1}{2}} d x$ is an antiderivative of $2 x {(7 + x^{2})}^{\frac{1}{2}}$ , so by definition $\frac{d}{d x} [\int 2 x {(7 + x^{2})}^{\frac{1}{2}} d x]$ is $2 x {(7 + x^{2})}^{\frac{1}{2}}$ .

$2 x {(7 + x^{2})}^{\frac{1}{2}}$