Calculus Examples

$\int \frac{x}{\sqrt{a + b x}} d x$

Step 1

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

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

Step 2

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

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