Calculus Examples

$f (x) = \tan (x) (4 \cos (x) + 8 \sec (x))$

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.

$F (x) = \int \tan (x) (4 \cos (x) + 8 \sec (x)) d x$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\int \tan (x) (4 \cos (x)) + \tan (x) (8 \sec (x)) d x$

Step 3.2

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.2.1

Reorder $\tan (x)$ and $4 \cos (x)$ .

$\int 4 \cos (x) \tan (x) + \tan (x) (8 \sec (x)) d x$

Step 3.2.2

Add parentheses.

$\int 4 (\cos (x) \tan (x)) + \tan (x) (8 \sec (x)) d x$

Step 3.2.3

Reorder $\cos (x)$ and $\tan (x)$ .

$\int 4 (\tan (x) \cos (x)) + \tan (x) (8 \sec (x)) d x$

Step 3.2.4

Rewrite $\tan (x) (4 \cos (x))$ in terms of sines and cosines.

$\int 4 (\frac{\sin (x)}{\cos (x)} \cos (x)) + \tan (x) (8 \sec (x)) d x$

Step 3.2.5

Cancel the common factors.

$\int 4 \sin (x) + \tan (x) (8 \sec (x)) d x$

Step 3.3

Rewrite using the commutative property of multiplication.

$\int 4 \sin (x) + 8 \tan (x) \sec (x) d x$

Step 4

Split the single integral into multiple integrals.

$\int 4 \sin (x) d x + \int 8 \tan (x) \sec (x) d x$

Step 5

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int \sin (x) d x + \int 8 \tan (x) \sec (x) d x$

Step 6

The integral of $\sin (x)$ with respect to $x$ is $- \cos (x)$ .

$4 (- \cos (x) + C) + \int 8 \tan (x) \sec (x) d x$

Step 7

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$4 (- \cos (x) + C) + 8 \int \tan (x) \sec (x) d x$

Step 8

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

$4 (- \cos (x) + C) + 8 (\sec (x) + C)$

Step 9

Simplify.

$- 4 \cos (x) + 8 \sec (x) + C$

Step 10

The answer is the antiderivative of the function $f (x) = \tan (x) (4 \cos (x) + 8 \sec (x))$ .

$F (x) =$ $- 4 \cos (x) + 8 \sec (x) + C$