Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 4

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

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

Step 5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 6

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

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

Step 7

Simplify.

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