Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.2.1

Reorder $\sin (x)$ and $\cot (x)$ .

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

Step 1.2.2

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

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

Step 1.2.3

Cancel the common factors.

$\int \cos (x) + \sin (x) \csc (x) d x$

Step 1.3

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.3.1

Reorder $\sin (x)$ and $\csc (x)$ .

$\int \cos (x) + \csc (x) \sin (x) d x$

Step 1.3.2

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

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

Step 1.3.3

Cancel the common factors.

$\int \cos (x) + 1 d x$

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

$\sin (x) + C + \int d x$

Step 4

Apply the constant rule.

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

Step 5

Simplify.

$\sin (x) + x + C$