Calculus Examples

$\int \frac{\sin (x)}{\cos^{3} (x)} d x$

Step 1

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

$\frac{\sin (x)}{\cos^{3} (x)}$

Step 2

Simplify.

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Step 2.1

Factor $\cos (x)$ out of $\cos^{3} (x)$ .

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

Step 2.2

Separate fractions.

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

Step 2.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Step 2.4

Rewrite $1$ as $1^{2}$ .

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

Step 2.5

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

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

Step 2.6

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec^{2} (x) \tan (x)$