Calculus Examples

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

Step 1

Write $\sec (x) (\tan (x) - \sec (x))$ as a function.

$f (x) = \sec (x) (\tan (x) - \sec (x))$

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.

$F (x) = \int \sec (x) (\tan (x) - \sec (x)) d x$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Rewrite using the commutative property of multiplication.

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

Step 4.3

Multiply $- \sec (x) \sec (x)$ .

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Step 4.3.1

Raise $\sec (x)$ to the power of $1$ .

$\int \sec (x) \tan (x) - (\sec^{1} (x) \sec (x)) d x$

Step 4.3.2

Raise $\sec (x)$ to the power of $1$ .

$\int \sec (x) \tan (x) - (\sec^{1} (x) \sec^{1} (x)) d x$

Step 4.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \sec (x) \tan (x) - {\sec (x)}^{1 + 1} d x$

Step 4.3.4

Add $1$ and $1$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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

Step 10

The answer is the antiderivative of the function $f (x) = \sec (x) (\tan (x) - \sec (x))$ .

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