Calculus Examples

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

Step 1

Since

$\frac{x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)}$ is constant with respect to

$d$ , move

$\frac{x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)}$ out of the integral.

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

Step 2

By the Power Rule, the integral of

$d$ with respect to

$d$ is

$\frac{1}{2} d^{2}$ .

$\frac{x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)} (\frac{1}{2} d^{2} + C)$

Step 3

Simplify the answer.

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Step 3.1

Rewrite

$\frac{x}{(2 + \cos (x) - 2 \sin (x)) \sin (x)} (\frac{1}{2} d^{2} + C)$ as

$\frac{x}{2 + \cos (x) - 2 \sin (x)} \csc (x) (\frac{1}{2} d^{2}) + C$ .

$\frac{x}{2 + \cos (x) - 2 \sin (x)} \csc (x) (\frac{1}{2} d^{2}) + C$

Step 3.2

Simplify.

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Step 3.2.1

Combine

$\frac{x}{2 + \cos (x) - 2 \sin (x)}$ and

$\csc (x)$ .

$\frac{x \csc (x)}{2 + \cos (x) - 2 \sin (x)} (\frac{1}{2} d^{2}) + C$

Step 3.2.2

Combine

$\frac{1}{2}$ and

$d^{2}$ .

$\frac{x \csc (x)}{2 + \cos (x) - 2 \sin (x)} \cdot \frac{d^{2}}{2} + C$

Step 3.2.3

Multiply

$\frac{x \csc (x)}{2 + \cos (x) - 2 \sin (x)}$ by

$\frac{d^{2}}{2}$ .

$\frac{x \csc (x) d^{2}}{(2 + \cos (x) - 2 \sin (x)) \cdot 2} + C$

Step 3.2.4

Move

$2$ to the left of

$2 + \cos (x) - 2 \sin (x)$ .

$\frac{x \csc (x) d^{2}}{2 (2 + \cos (x) - 2 \sin (x))} + C$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Rewrite

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

$\frac{x \frac{1}{\sin (x)} d^{2}}{2 (2 + \cos (x) - 2 \sin (x))} + C$

Step 3.3.1.2

Combine exponents.

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Step 3.3.1.2.1

Combine

$x$ and

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

$\frac{\frac{x}{\sin (x)} d^{2}}{2 (2 + \cos (x) - 2 \sin (x))} + C$

Step 3.3.1.2.2

Combine

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

$d^{2}$ .

$\frac{\frac{x d^{2}}{\sin (x)}}{2 (2 + \cos (x) - 2 \sin (x))} + C$

Step 3.3.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{x d^{2}}{\sin (x)} \cdot \frac{1}{2 (2 + \cos (x) - 2 \sin (x))} + C$

Step 3.3.3

Combine.

$\frac{x d^{2} \cdot 1}{\sin (x) (2 (2 + \cos (x) - 2 \sin (x)))} + C$

Step 3.3.4

Multiply

$x$ by

$1$ .

$\frac{d^{2} \cdot x}{\sin (x) \cdot 2 (2 + \cos (x) - 2 \sin (x))} + C$

Step 3.3.5

Move

$2$ to the left of

$\sin (x)$ .

$\frac{d^{2} x}{2 \sin (x) (2 + \cos (x) - 2 \sin (x))} + C$