## Problem Statement

Calculate the indefinite integral:

$$
\int(x+5) \sin 3 x \, dx
$$
## Solution

$$
\int(x+5) \sin 3 x d x=
$$

Let's denote:

$$
\begin{aligned}
& u=x+5 ; d u=d x \\
& d v=\sin 3 x d x ; v=-\frac{1}{3} \cos 3 x
\end{aligned}
$$

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

$$
\begin{aligned}
& =(x+5) \cdot\left(-\frac{1}{3} \cos 3 x\right)-\int\left(-\frac{1}{3} \cos 3 x\right) d x= \\
& =-\frac{1}{3} \cdot(x+5) \cos 3 x+\frac{1}{3} \int \cos 3 x d x=-\frac{1}{3} \cdot(x+5) \cos 3 x+\frac{1}{3} \cdot \frac{1}{3} \cdot \sin 3 x+C= \\
& =-\frac{1}{3} \cdot(x+5) \cos 3 x+\frac{1}{9} \cdot \sin 3 x+C
\end{aligned}
$$

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%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D1%8B_1-23" Categories: Kuznetsov's Problem Book Integrals Problem 1 | Integrals

Ukrainian Banner Network

- Last modified on this page: 13:05, 20 February 2009.
- Content is available under CC-BY-SA 3.0.

Created by GeeTeatoo

## Problem Kuznetsov Integrals $1-24$

## Material from PlusPi