Calculus Examples

$f (x) = \frac{1}{\cos^{2} (x)} - \frac{2}{3}$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int \frac{1}{\cos^{2} (x)} - \frac{2}{3} d x$

Step 3

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

$\int \sec^{2} (x) - \frac{2}{3} d x$

Step 4

Split the single integral into multiple integrals.

$\int \sec^{2} (x) d x + \int - \frac{2}{3} d x$

Step 5

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

$\tan (x) + C + \int - \frac{2}{3} d x$

Step 6

Apply the constant rule.

$\tan (x) + C - \frac{2}{3} x + C$

Step 7

Simplify.

$\tan (x) - \frac{2}{3} x + C$

Step 8

The answer is the antiderivative of the function $f (x) = \frac{1}{\cos^{2} (x)} - \frac{2}{3}$ .

$F (x) =$ $\tan (x) - \frac{2}{3} x + C$