## Problem Statement

Calculate the indefinite integral:

$$
\int \frac{x}{\sin ^{2} x} d x
$$
## Solution

$$
\int \frac{x}{\sin ^{2} x} d x=
$$

Let's denote:

$$
\begin{aligned}
& u=x ; d u=d x \\
& d v=\frac{1}{\sin ^{2} x} d x ; v=-\cot x
\end{aligned}
$$

We will use the integration by parts formula $\int u d v=u v-\int v d u$. We get:

$$
\begin{aligned}
& =x \cdot(-\cot x)-\int(-\cot x) d x=-x \cdot \cot x+\int \frac{\cos x}{\sin x} d x= \\
& =-x \cdot \cot x+\int \frac{d(\sin x)}{\sin x}=-x \cdot \cot x+\ln |\sin x|+C
\end{aligned}
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� \%D0\%9A\%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+1-29$ »

Categories: Kuznetsov's Problem Book Integrals Problem 1 | Integrals

- Last modified: 16:28, 20 February 2009.
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Created by GeeTeatoo

## Problem Kuznetsov Integrals 1-30

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