Calculus Examples

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

Step 1

Write $\frac{\sin (2 x)}{\sin (x)}$ as a function.

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

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.

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

Step 4

Apply the sine double-angle identity.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

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

$\int 2 \cos (x) d x$

Step 6

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int \cos (x) d x$

Step 7

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

$2 (\sin (x) + C)$

Step 8

Simplify.

$2 \sin (x) + C$

Step 9

The answer is the antiderivative of the function $f (x) = \frac{\sin (2 x)}{\sin (x)}$ .

$F (x) =$ $2 \sin (x) + C$