Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply the constant rule.

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

Step 6

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

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

Step 7

Apply the constant rule.

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

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

Add $- x$ and $x$ .

$C - \cot (x) + C + 0 + C$

Step 8.1.2

Add $C - \cot (x) + C$ and $0$ .

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

Step 8.2

Simplify.

$- \cot (x) + C$