## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{2 \pi}\left(1-8 x^{2}\right) \cos 4 x \, d x
$$
## Solution

$$
\int_{0}^{2 \pi}\left(1-8 x^{2}\right) \cos 4 x d x=
$$

Let's denote:

$$
\begin{aligned}
& u=1-8 x^{2} ; d u=-16 x \cdot d x \\
& d v=\cos 4 x d x ; v=\frac{1}{4} \sin 4 x
\end{aligned}
$$

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

$$
\begin{aligned}
& =\left.\left(1-8 x^{2}\right) \cdot \frac{1}{4} \sin 4 x\right|_{0} ^{2 \pi}-\int_{0}^{2 \pi} \frac{1}{4} \sin 4 x \cdot(-16 x) \cdot d x= \\
& =\left(1-8 \cdot(2 \pi)^{2}\right) \cdot \frac{1}{4} \sin (4 \cdot 2 \pi)-\left(1-8 \cdot 0^{2}\right) \cdot \frac{1}{4} \sin (4 \cdot 0)+4 \int_{0}^{2 \pi} \sin 4 x \cdot x \cdot d x= \\
& =\left(1-32 \pi^{2}\right) \cdot \frac{1}{4} \cdot 0-1 \cdot \frac{1}{4} \cdot 0+4 \int_{0}^{2 \pi} \sin 4 x \cdot x \cdot d x=4 \int_{0}^{2 \pi} \sin 4 x \cdot x \cdot d x=
\end{aligned}
$$

Let's denote:

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

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

$$
\begin{aligned}
& =4 \cdot\left(\left.x \cdot\left(-\frac{1}{4} \cos 4 x\right)\right|_{0} ^{2 \pi}-\int_{0}^{2 \pi}\left(-\frac{1}{4} \cos 4 x\right) d x\right)= \\
& =4 \cdot\left(2 \pi \cdot\left(-\frac{1}{4} \cos (4 \cdot 2 \pi)\right)-0 \cdot\left(-\frac{1}{2} \cos (4 \cdot 0)\right)+\frac{1}{4} \int_{0}^{2 \pi} \cos 4 x d x\right)= \\
& =4 \cdot\left(-\frac{\pi}{2}+\left.\frac{1}{4} \cdot \frac{1}{4} \sin 4 x\right|_{0} ^{2 \pi}\right)=4 \cdot\left(-\frac{\pi}{2}+\frac{1}{16} \sin (4 \cdot 2 \pi)-\frac{1}{4} \sin (4 \cdot 0)\right)= \\
& =4 \cdot\left(-\frac{\pi}{2}+0-0\right)=-2 \pi
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}$ 2-12»

Categories: Kuznetsov's Problem Book Integrals Problem 2 | Integrals

Ukrainian Banner Network

- Last modified: 21:11, 25 February 2009.
- Content is available under CC-BY-SA 3.0.

Created by GeeTeatoo

## Problem Kuznetsov Integrals 2-13

## Material from PlusPi