## Problem Statement

Calculate the indefinite integral:

$$
\int(2 x-5) \cos 4 x \, d x
$$
## Solution

$$
\int(2 x-5) \cos 4 x d x=
$$

Let's denote:

$$
\begin{aligned}
& u=2 x-5 ; d u=2 d x \\
& d v=\cos 4 x d x ; v=\frac{1}{4} \sin 4 x
\end{aligned}
$$

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

$$
\begin{aligned}
& =(2 x-5) \cdot \frac{1}{4} \sin 4 x-\int \frac{1}{4} \sin 4 x \cdot 2 d x=\frac{1}{4}(2 x-5) \sin 4 x-\frac{1}{2} \int \sin 4 x d x= \\
& =\frac{1}{4}(2 x-5) \sin 4 x+\frac{1}{2} \cdot \frac{1}{4} \cdot \cos 4 x+C=\frac{1}{4}(2 x-5) \sin 4 x+\frac{1}{8} \cdot \cos 4 x+C
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+1-21$ » Categories: Kuznetsov's Problem Book Integrals Problem 1 | Integrals

- Last edited: 12:53, 20 February 2009.
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Created by GeeTeatoo

## Problem Kuznetsov Integrals $1-22$

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