Calculus Examples

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

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.

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Convert from $\frac{1}{\sin^{2} (x)}$ to $\csc^{2} (x)$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

$\tan (x) + C + \int - \csc^{2} (x) d x$

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\tan (x) + C - \int \csc^{2} (x) d x$

Step 7

Since the derivative of $- \cot (x)$ is $\csc^{2} (x)$ , the integral of $\csc^{2} (x)$ is $- \cot (x)$ .

$\tan (x) + C - (- \cot (x) + C)$

Step 8

Simplify.

$\tan (x) + \cot (x) + C$

Step 9

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

$F (x) =$ $\tan (x) + \cot (x) + C$