Calculus Examples

$\int \sin (x) \sec^{2} (x) d x$

Step 1

Simplify $\sin (x) \sec^{2} (x)$ .

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Step 1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 1.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 1.3

One to any power is one.

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

Step 1.4

Combine $\sin (x)$ and $\frac{1}{\cos^{2} (x)}$ .

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

Step 1.5

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 1.6

Separate fractions.

$\int \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} d x$

Step 1.7

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

$\int \frac{1}{\cos (x)} \tan (x) d x$

Step 1.8

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

$\int \sec (x) \tan (x) d x$

Step 2

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

$\sec (x) + C$