Calculus Examples

$\frac{x}{\cos^{2} (x)}$

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

Multiply by $1$ .

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

Step 4.2

Separate fractions.

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

Step 4.3

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

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

Step 4.4

Divide $x$ by $1$ .

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

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sec^{2} (x)$ .

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

Step 6

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

$x \tan (x) - (\ln (| \sec (x) |) + C)$

Step 7

Simplify.

$x \tan (x) - \ln (| \sec (x) |) + C$

Step 8

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

$F (x) =$ $x \tan (x) - \ln (| \sec (x) |) + C$