Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 1.2

Apply the distributive property.

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

Cancel the common factors.

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

Step 1.6

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

$\ln (| \sec (x) |) + C + \int d x$

Step 4

Apply the constant rule.

$\ln (| \sec (x) |) + C + x + C$

Step 5

Simplify.

$\ln (| \sec (x) |) + x + C$