Calculus Examples

$\frac{2 π}{\cos^{2} (x)}$

Step 1

Write $\frac{2 π}{\cos^{2} (x)}$ as a function.

$f (x) = \frac{2 π}{\cos^{2} (x)}$

Step 2

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 3

Set up the integral to solve.

$F (x) = \int \frac{2 π}{\cos^{2} (x)} d x$

Step 4

Since $2 π$ is constant with respect to $x$ , move $2 π$ out of the integral.

$2 π \int \frac{1}{\cos^{2} (x)} d x$

Step 5

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

$2 π \int \sec^{2} (x) d x$

Step 6

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

$2 π (\tan (x) + C)$

Step 7

Simplify.

$2 π \tan (x) + C$

Step 8

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

$F (x) =$ $2 π \tan (x) + C$