Calculus Examples

$\int x^{3} \sqrt{x^{2} + 1} d x$

Step 1

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

$\frac{d}{d x} [\int x^{3} {(x^{2} + 1)}^{\frac{1}{2}} d x]$

Step 2

$\int x^{3} {(x^{2} + 1)}^{\frac{1}{2}} d x$ is an antiderivative of $x^{3} {(x^{2} + 1)}^{\frac{1}{2}}$ , so by definition $\frac{d}{d x} [\int x^{3} {(x^{2} + 1)}^{\frac{1}{2}} d x]$ is $x^{3} {(x^{2} + 1)}^{\frac{1}{2}}$ .

$x^{3} {(x^{2} + 1)}^{\frac{1}{2}}$