Calculus Examples

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

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

Step 1

Since

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

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

d

$d$ , move

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

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

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

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

Step 2

By the Power Rule, the integral of

d

$d$ with respect to

d

$d$ is

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

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

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

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

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)

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

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

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

\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)}

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

\csc (x)

$\csc (x)$ .

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

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

Step 3.2.2

Combine

\frac{1}{2}

$\frac{1}{2}$ and

d^{2}

$d^{2}$ .

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

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

Step 3.2.3

Multiply

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

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

\frac{d^{2}}{2}

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

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

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

Step 3.2.4

Move

2

$2$ to the left of

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

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

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

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

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

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

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)

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

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

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

Step 3.3.1.2

Combine exponents.

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Step 3.3.1.2.1

Combine

x

$x$ and

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

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

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

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

Step 3.3.1.2.2

Combine

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

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

d^{2}

$d^{2}$ .

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

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

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

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

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

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

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

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

Step 3.3.3

Combine.

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

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

Step 3.3.4

Multiply

x

$x$ by

1

$1$ .

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

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

Step 3.3.5

Move

2

$2$ to the left of

\sin (x)

$\sin (x)$ .

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

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

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

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

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

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