Calculus Examples

$f (x) = \cot (x) (- 8 \sin (x) - 10 \csc (x))$

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.

$F (x) = \int \cot (x) (- 8 \sin (x) - 10 \csc (x)) d x$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\int \cot (x) (- 8 \sin (x)) + \cot (x) (- 10 \csc (x)) d x$

Step 3.2

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

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Step 3.2.1

Reorder $\cot (x)$ and $- 8 \sin (x)$ .

$\int - 8 \sin (x) \cot (x) + \cot (x) (- 10 \csc (x)) d x$

Step 3.2.2

Add parentheses.

$\int - 8 (\sin (x) \cot (x)) + \cot (x) (- 10 \csc (x)) d x$

Step 3.2.3

Reorder $\sin (x)$ and $\cot (x)$ .

$\int - 8 (\cot (x) \sin (x)) + \cot (x) (- 10 \csc (x)) d x$

Step 3.2.4

Rewrite $\cot (x) (- 8 \sin (x))$ in terms of sines and cosines.

$\int - 8 (\frac{\cos (x)}{\sin (x)} \sin (x)) + \cot (x) (- 10 \csc (x)) d x$

Step 3.2.5

Cancel the common factors.

$\int - 8 \cos (x) + \cot (x) (- 10 \csc (x)) d x$

Step 3.3

Rewrite using the commutative property of multiplication.

$\int - 8 \cos (x) - 10 \cot (x) \csc (x) d x$

Step 4

Split the single integral into multiple integrals.

$\int - 8 \cos (x) d x + \int - 10 \cot (x) \csc (x) d x$

Step 5

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

$- 8 \int \cos (x) d x + \int - 10 \cot (x) \csc (x) d x$

Step 6

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

$- 8 (\sin (x) + C) + \int - 10 \cot (x) \csc (x) d x$

Step 7

Since $- 10$ is constant with respect to $x$ , move $- 10$ out of the integral.

$- 8 (\sin (x) + C) - 10 \int \cot (x) \csc (x) d x$

Step 8

Since the derivative of $- \csc (x)$ is $\cot (x) \csc (x)$ , the integral of $\cot (x) \csc (x)$ is $- \csc (x)$ .

$- 8 (\sin (x) + C) - 10 (- \csc (x) + C)$

Step 9

Simplify.

$- 8 \sin (x) + 10 \csc (x) + C$

Step 10

The answer is the antiderivative of the function $f (x) = \cot (x) (- 8 \sin (x) - 10 \csc (x))$ .

$F (x) =$ $- 8 \sin (x) + 10 \csc (x) + C$