## Problem Statement

Calculate the indefinite integral:

$$
\int(8-3 x) \cos 5 x \, dx
$$
## Solution

$$
\int(8-3 x) \cos 5 x d x=
$$

Let's denote:

$$
\begin{aligned}
& u=8-3 x ; d u=-3 d x \\
& d v=\cos 5 x d x ; v=\frac{1}{5} \sin 5 x
\end{aligned}
$$

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

$$
\begin{aligned}
& =(8-3 x) \cdot \frac{1}{5} \sin 5 x-\int \frac{1}{5} \sin 5 x \cdot(-3) d x=\frac{1}{5}(8-3 x) \sin 5 x+\frac{3}{5} \int \sin 5 x d x= \\
& =\frac{1}{5}(8-3 x) \sin 5 x-\frac{3}{5} \cdot \frac{1}{5} \cdot \cos 5 x+C=\frac{1}{5}(8-3 x) \sin 5 x-\frac{3}{25} \cdot \cos 5 x+C
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+1-22$ »

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

- Last modified: 12:57, 20 February 2009.
- Content is available under CC-BY-SA 3.0.

Created by GeeTeatoo

## Problem Kuznetsov Integrals $1-23$

## Material from Plusi