Calculus Examples

$\frac{1}{\sec (x) \tan (x)}$

Step 1

Write $\frac{1}{\sec (x) \tan (x)}$ as a function.

$f (x) = \frac{1}{\sec (x) \tan (x)}$

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.

$F (x) = \int \frac{1}{\sec (x) \tan (x)} d x$

Step 4

Let $u = \sec (x)$ . Then $d u = \sec (x) \tan (x) d x$ , so $- d u = - \sec (x) \tan (x) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = \sec (x)$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $\sec (x)$ .

$\frac{d}{d x} [\sec (x)]$

Step 4.1.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\sec (x) \tan (x)$

Step 4.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{1} \cdot 1 d u$

Step 5

Simplify.

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Step 5.1

Cancel the common factor of $1$ .

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Step 5.1.1

Cancel the common factor.

$\int \frac{1}{1} \cdot 1 d u$

Step 5.1.2

Rewrite the expression.

$\int 1 \cdot 1 d u$

Step 5.2

Multiply $1$ by $1$ .

$\int d u$

Step 6

Apply the constant rule.

$u + C$

Step 7

Replace all occurrences of $u$ with $\sec (x)$ .

$\sec (x) + C$

Step 8

The answer is the antiderivative of the function $f (x) = \frac{1}{\sec (x) \tan (x)}$ .

$F (x) =$ $\sec (x) + C$