Step-by-step Solution

Solve the trigonometric integral $\int x\csc\left(x\right)^2dx$

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$-x\cot\left(x\right)+\ln\left|\sin\left(x\right)\right|+C_0$

Step-by-step explanation

Problem to solve:

$\int\left(x\cdot\csc\left(x\right)^2\right)dx$
1

Use the integration by parts theorem to calculate the integral $\int x\csc\left(x\right)^2dx$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
2

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=x}\\ \displaystyle{du=dx}\end{matrix}$

$-x\cot\left(x\right)+\ln\left|\sin\left(x\right)\right|+C_0$
$\int\left(x\cdot\csc\left(x\right)^2\right)dx$