Calculus Examples

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

Step 1

Rewrite $\frac{\sin (2 x)}{\sin (x)}$ as a product.

$\frac{d}{d x} [\int \sin (2 x) \frac{1}{\sin (x)} d x]$

Step 2

Write $\sin (2 x)$ as a fraction with denominator $1$ .

$\frac{d}{d x} [\int \frac{\sin (2 x)}{1} \cdot \frac{1}{\sin (x)} d x]$

Step 3

Simplify.

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Step 3.1

Divide $\sin (2 x)$ by $1$ .

$\frac{d}{d x} [\int \sin (2 x) \frac{1}{\sin (x)} d x]$

Step 3.2

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

$\frac{d}{d x} [\int \sin (2 x) \csc (x) d x]$

Step 4

$\int \sin (2 x) \csc (x) d x$ is an antiderivative of $\sin (2 x) \csc (x)$ , so by definition $\frac{d}{d x} [\int \sin (2 x) \csc (x) d x]$ is $\sin (2 x) \csc (x)$ .

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Step 4.1

Divide $\sin (2 x)$ by $1$ .

$\sin (2 x) \frac{1}{\sin (x)}$

Step 4.2

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

$\sin (2 x) \csc (x)$

Step 5

Simplify.

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Step 5.1

Reorder the factors of $\sin (2 x) \csc (x)$ .

$\csc (x) \sin (2 x)$

Step 5.2

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

$\frac{1}{\sin (x)} \sin (2 x)$

Step 5.3

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

$\frac{\sin (2 x)}{\sin (x)}$

Step 5.4

Apply the sine double-angle identity.

$\frac{2 \sin (x) \cos (x)}{\sin (x)}$

Step 5.5

Cancel the common factor of $\sin (x)$ .

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Step 5.5.1

Cancel the common factor.

$\frac{2 \sin (x) \cos (x)}{\sin (x)}$

Step 5.5.2

Divide $2 \cos (x)$ by $1$ .

$2 \cos (x)$