Calculus Examples

$\int (7 \csc^{2} (x) + 2 \sec (x) \tan (x)) d x$

Step 1

Remove parentheses.

$\int 7 \csc^{2} (x) + 2 \sec (x) \tan (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int 7 \csc^{2} (x) d x + \int 2 \sec (x) \tan (x) d x$

Step 3

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int \csc^{2} (x) d x + \int 2 \sec (x) \tan (x) d x$

Step 4

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

$7 (- \cot (x) + C) + \int 2 \sec (x) \tan (x) d x$

Step 5

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

$7 (- \cot (x) + C) + 2 \int \sec (x) \tan (x) d x$

Step 6

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

$7 (- \cot (x) + C) + 2 (\sec (x) + C)$

Step 7

Simplify.

$- 7 \cot (x) + 2 \sec (x) + C$