## Problem Statement

Calculate the indefinite integral:

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

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

Let's denote:

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

Using the integration by parts formula $\int u d v=u v-\int v d u$, we get:

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

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+1-4$ »

Categories: Kuznetsov's Problem Book Integrals Problem 1 | Integrals | Problems for Checking

- Last edited on this page: 08:55, 9 July 2009.
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## Problem Kuznetsov Integrals $1-5$

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