Calculus Examples

$\int \frac{1 + \tan^{2} (x)}{\tan (x)} d x$

Step 1

Split the fraction $\frac{1 + \tan^{2} (x)}{\tan (x)}$ into two fractions.

$\int \frac{1}{\tan (x)} + \frac{\tan^{2} (x)}{\tan (x)} d x$

Step 2

Simplify.

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Step 2.1

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

$\int \frac{1}{\frac{\sin (x)}{\cos (x)}} + \frac{\tan^{2} (x)}{\tan (x)} d x$

Step 2.2

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 2.3

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

$\int \cot (x) + \frac{\tan^{2} (x)}{\tan (x)} d x$

Step 2.4

Cancel the common factor of $\tan^{2} (x)$ and $\tan (x)$ .

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Step 2.4.1

Factor $\tan (x)$ out of $\tan^{2} (x)$ .

$\int \cot (x) + \frac{\tan (x) \tan (x)}{\tan (x)} d x$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Multiply by $1$ .

$\int \cot (x) + \frac{\tan (x) \tan (x)}{\tan (x) \cdot 1} d x$

Step 2.4.2.2

Cancel the common factor.

$\int \cot (x) + \frac{\tan (x) \tan (x)}{\tan (x) \cdot 1} d x$

Step 2.4.2.3

Rewrite the expression.

$\int \cot (x) + \frac{\tan (x)}{1} d x$

Step 2.4.2.4

Divide $\tan (x)$ by $1$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

The integral of $\cot (x)$ with respect to $x$ is $\ln (| \sin (x) |)$ .

$\ln (| \sin (x) |) + C + \int \tan (x) d x$

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Simplify.

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

Step 6.2

Simplify.

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Step 6.2.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 6.2.2

To multiply absolute values, multiply the terms inside each absolute value.

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