Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \csc^{2} (x)$ .

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

Step 2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 3

Simplify.

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Step 3.1

Multiply $- 1$ by $- 1$ .

$x (- \cot (x)) + 1 \int \cot (x) d x$

Step 3.2

Multiply $\int \cot (x) d x$ by $1$ .

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

Step 4

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

$x (- \cot (x)) + \ln (| \sin (x) |) + C$

Step 5

Rewrite $x (- \cot (x)) + \ln (| \sin (x) |) + C$ as $- x \cot (x) + \ln (| \sin (x) |) + C$ .

$- x \cot (x) + \ln (| \sin (x) |) + C$