Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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Step 4.2.1

Rewrite $\cot (x) \sec (x)$ in terms of sines and cosines.

$\int \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos (x)} + \cot (x) (- \tan (x)) d x$

Step 4.2.2

Cancel the common factors.

$\int \frac{1}{\sin (x)} + \cot (x) (- \tan (x)) d x$

Step 4.3

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

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Step 4.3.1

Reorder $\cot (x)$ and $- \tan (x)$ .

$\int \frac{1}{\sin (x)} - \tan (x) \cot (x) d x$

Step 4.3.2

Add parentheses.

$\int \frac{1}{\sin (x)} - (\tan (x) \cot (x)) d x$

Step 4.3.3

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

$\int \frac{1}{\sin (x)} - (\cot (x) \tan (x)) d x$

Step 4.3.4

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

$\int \frac{1}{\sin (x)} - (\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{\cos (x)}) d x$

Step 4.3.5

Cancel the common factors.

$\int \frac{1}{\sin (x)} - 1 \cdot 1 d x$

Step 4.4

Simplify each term.

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Step 4.4.1

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\int \csc (x) - 1 \cdot 1 d x$

Step 4.4.2

Multiply $- 1$ by $1$ .

$\int \csc (x) - 1 d x$

Step 5

Split the single integral into multiple integrals.

$\int \csc (x) d x + \int - 1 d x$

Step 6

The integral of $\csc (x)$ with respect to $x$ is $\ln (| \csc (x) - \cot (x) |)$ .

$\ln (| \csc (x) - \cot (x) |) + C + \int - 1 d x$

Step 7

Apply the constant rule.

$\ln (| \csc (x) - \cot (x) |) + C - x + C$

Step 8

Simplify.

$\ln (| \csc (x) - \cot (x) |) - x + C$

Step 9

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

$F (x) =$ $\ln (| \csc (x) - \cot (x) |) - x + C$