Calculus Examples

$\int \frac{\csc (x) \cos (x)}{\sin (x)} d x$

Step 1

Separate fractions.

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

Step 2

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

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

Step 3

Rewrite $\frac{\frac{1}{\sin (x)}}{\sin (x)}$ as a product.

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

Step 4

Combine fractions.

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Step 4.1

Multiply $\frac{1}{\sin (x)}$ by $\frac{1}{\sin (x)}$ .

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

Step 4.2

Divide $\cos (x)$ by $1$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 5.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.4

Add $1$ and $1$ .

$\int \cos (x) \frac{1}{\sin^{2} (x)} d x$

Step 6

Simplify terms.

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Step 6.1

Combine $\cos (x)$ and $\frac{1}{\sin^{2} (x)}$ .

$\int \frac{\cos (x)}{\sin^{2} (x)} d x$

Step 6.2

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

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

Step 7

Separate fractions.

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

Step 8

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 9

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

$\int \csc (x) \cot (x) d x$

Step 10

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

$- \csc (x) + C$