Calculus Examples

$\int \tan (x) (6 \cos (x) - 6 \sec (x)) d x$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int \tan (x) (6 \cos (x)) + \tan (x) (- 6 \sec (x)) d x$

Step 1.2

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

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Step 1.2.1

Reorder $\tan (x)$ and $6 \cos (x)$ .

$\int 6 \cos (x) \tan (x) + \tan (x) (- 6 \sec (x)) d x$

Step 1.2.2

Add parentheses.

$\int 6 (\cos (x) \tan (x)) + \tan (x) (- 6 \sec (x)) d x$

Step 1.2.3

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

$\int 6 (\tan (x) \cos (x)) + \tan (x) (- 6 \sec (x)) d x$

Step 1.2.4

Rewrite $\tan (x) (6 \cos (x))$ in terms of sines and cosines.

$\int 6 (\frac{\sin (x)}{\cos (x)} \cos (x)) + \tan (x) (- 6 \sec (x)) d x$

Step 1.2.5

Cancel the common factors.

$\int 6 \sin (x) + \tan (x) (- 6 \sec (x)) d x$

Step 1.3

Rewrite using the commutative property of multiplication.

$\int 6 \sin (x) - 6 \tan (x) \sec (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int 6 \sin (x) d x + \int - 6 \tan (x) \sec (x) d x$

Step 3

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int \sin (x) d x + \int - 6 \tan (x) \sec (x) d x$

Step 4

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

$6 (- \cos (x) + C) + \int - 6 \tan (x) \sec (x) d x$

Step 5

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$6 (- \cos (x) + C) - 6 \int \tan (x) \sec (x) d x$

Step 6

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

$6 (- \cos (x) + C) - 6 (\sec (x) + C)$

Step 7

Simplify.

$- 6 \cos (x) - 6 \sec (x) + C$