Calculus Examples

$\int 4 \cos (x) (4 \tan (x) + 3 \sec (x)) d x$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int 4 \cos (x) (4 \tan (x)) + 4 \cos (x) (3 \sec (x)) d x$

Step 1.2

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

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Step 1.2.1

Add parentheses.

$\int 4 (\cos (x) (4 \tan (x))) + 4 \cos (x) (3 \sec (x)) d x$

Step 1.2.2

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

$\int 4 (4 \tan (x) \cos (x)) + 4 \cos (x) (3 \sec (x)) d x$

Step 1.2.3

Add parentheses.

$\int 4 (4 (\tan (x) \cos (x))) + 4 \cos (x) (3 \sec (x)) d x$

Step 1.2.4

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

$\int 4 (4 (\frac{\sin (x)}{\cos (x)} \cos (x))) + 4 \cos (x) (3 \sec (x)) d x$

Step 1.2.5

Cancel the common factors.

$\int 4 (4 \sin (x)) + 4 \cos (x) (3 \sec (x)) d x$

Step 1.3

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

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Step 1.3.1

Add parentheses.

$\int 4 (4 \sin (x)) + 4 (\cos (x) (3 \sec (x))) d x$

Step 1.3.2

Reorder $\cos (x)$ and $3 \sec (x)$ .

$\int 4 (4 \sin (x)) + 4 (3 \sec (x) \cos (x)) d x$

Step 1.3.3

Add parentheses.

$\int 4 (4 \sin (x)) + 4 (3 (\sec (x) \cos (x))) d x$

Step 1.3.4

Rewrite $4 \cos (x) (3 \sec (x))$ in terms of sines and cosines.

$\int 4 (4 \sin (x)) + 4 (3 (\frac{1}{\cos (x)} \cos (x))) d x$

Step 1.3.5

Cancel the common factors.

$\int 4 (4 \sin (x)) + 4 (3 \cdot 1) d x$

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $4$ by $4$ .

$\int 16 \sin (x) + 4 (3 \cdot 1) d x$

Step 1.4.2

Multiply $4 (3 \cdot 1)$ .

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Step 1.4.2.1

Multiply $3$ by $1$ .

$\int 16 \sin (x) + 4 \cdot 3 d x$

Step 1.4.2.2

Multiply $4$ by $3$ .

$\int 16 \sin (x) + 12 d x$

Step 2

Split the single integral into multiple integrals.

$\int 16 \sin (x) d x + \int 12 d x$

Step 3

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$16 \int \sin (x) d x + \int 12 d x$

Step 4

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

$16 (- \cos (x) + C) + \int 12 d x$

Step 5

Apply the constant rule.

$16 (- \cos (x) + C) + 12 x + C$

Step 6

Simplify.

$- 16 \cos (x) + 12 x + C$