Calculus Examples

$\int \frac{\cos (\sqrt{x})}{\sqrt{x}} d x$

Step 1

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

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

Step 2

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

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

Step 3

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

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