Calculus Examples

$\int (\tan^{2} (x) + \cot^{2} (x) + 4) d x$

Step 1

Remove parentheses.

$\int \tan^{2} (x) + \cot^{2} (x) + 4 d x$

Step 2

Split the single integral into multiple integrals.

$\int \tan^{2} (x) d x + \int \cot^{2} (x) d x + \int 4 d x$

Step 3

Using the Pythagorean Identity, rewrite $\tan^{2} (x)$ as $- 1 + \sec^{2} (x)$ .

$\int - 1 + \sec^{2} (x) d x + \int \cot^{2} (x) d x + \int 4 d x$

Step 4

Split the single integral into multiple integrals.

$\int - 1 d x + \int \sec^{2} (x) d x + \int \cot^{2} (x) d x + \int 4 d x$

Step 5

Apply the constant rule.

$- x + C + \int \sec^{2} (x) d x + \int \cot^{2} (x) d x + \int 4 d x$

Step 6

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

$- x + C + \tan (x) + C + \int \cot^{2} (x) d x + \int 4 d x$

Step 7

Using the Pythagorean Identity, rewrite $\cot^{2} (x)$ as $- 1 + \csc^{2} (x)$ .

$- x + C + \tan (x) + C + \int - 1 + \csc^{2} (x) d x + \int 4 d x$

Step 8

Split the single integral into multiple integrals.

$- x + C + \tan (x) + C + \int - 1 d x + \int \csc^{2} (x) d x + \int 4 d x$

Step 9

Apply the constant rule.

$- x + C + \tan (x) + C - x + C + \int \csc^{2} (x) d x + \int 4 d x$

Step 10

Since the derivative of $- \cot (x)$ is $\csc^{2} (x)$ , the integral of $\csc^{2} (x)$ is $- \cot (x)$ .

$- x + C + \tan (x) + C - x + C - \cot (x) + C + \int 4 d x$

Step 11

Apply the constant rule.

$- x + C + \tan (x) + C - x + C - \cot (x) + C + 4 x + C$

Step 12

Simplify.

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Step 12.1

Simplify.

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Step 12.1.1

Subtract $x$ from $- x$ .

$C + \tan (x) + C - 2 x + C - \cot (x) + C + 4 x + C$

Step 12.1.2

Add $- 2 x$ and $4 x$ .

$C + \tan (x) + C + C - \cot (x) + C + 2 x + C$

Step 12.2

Simplify.

$\tan (x) - \cot (x) + 2 x + C$