Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Multiply by $1$ .

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

Step 4

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

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

Step 5

Separate fractions.

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

Step 6

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

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

Step 7

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

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

Step 8

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

$\sec (x) + C + \int x d x$

Step 9

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

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

Step 10

Simplify.

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