Calculus Examples

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

Step 1

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

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

Step 2

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

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