## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{2 \pi}\left(2 x^{2}-15\right) \cos 3 x \, d x
$$
## Solution

$$
\int_{0}^{2 \pi}\left(2 x^{2}-15\right) \cos 3 x d x=
$$

Let:

$$
\begin{aligned}
& u=2 x^{2}-15 ; d u=4 x \cdot d x \\
& d v=\cos 3 x d x ; v=\frac{1}{3} \sin 3 x
\end{aligned}
$$

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

$$
\begin{aligned}
& =\left.\left(2 x^{2}-15\right) \cdot \frac{1}{3} \sin 3 x\right|_{0} ^{2 \pi}-\int_{0}^{2 \pi} \frac{1}{3} \sin 3 x \cdot 4 x \cdot d x= \\
& =\left(2 \pi^{2}-15\right) \cdot \frac{1}{3} \sin (3 \cdot 2 \pi)-\left(2 \cdot 0^{2}-15\right) \cdot \frac{1}{3} \sin (3 \cdot 0)-\frac{4}{3} \int_{0}^{2 \pi} \sin 3 x \cdot x \cdot d x= \\
& =\left(2 \pi^{2}-15\right) \cdot \frac{1}{3} \cdot 0+15 \cdot \frac{1}{3} \cdot 0-\frac{4}{3} \int_{0}^{2 \pi} \sin 3 x \cdot x \cdot d x=-\int_{0}^{2 \pi} \sin 3 x \cdot 4 x \cdot d x=
\end{aligned}
$$

Let:

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

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

$$
\begin{aligned}
& =-\frac{4}{3}\left(\left.x \cdot\left(-\frac{1}{3} \cos 3 x\right)\right|_{0} ^{2 \pi}-\int_{0}^{2 \pi}\left(-\frac{1}{3} \cos 3 x\right) \cdot d x\right)= \\
& =-\frac{4}{3}\left(2 \pi \cdot\left(-\frac{1}{3} \cos (3 \cdot \pi)\right)-0 \cdot\left(-\frac{1}{3} \cos (3 \cdot 0)\right)+\frac{1}{3} \cdot \int_{0}^{2 \pi} \cos 3 x d x\right)= \\
& =-\frac{4}{3}\left(-\frac{2 \pi}{3}+\left.\frac{1}{3} \cdot \frac{1}{3} \sin 3 x\right|_{0} ^{2 \pi}\right)=-\frac{4}{3}\left(-\frac{2 \pi}{3}+\frac{4}{9} \sin (3 \cdot 2 \pi)-\frac{1}{9} \sin (3 \cdot 0)\right)= \\
& =-\frac{4}{3}\left(-\frac{2 \pi}{3}+0-0\right)=\frac{8 \pi}{9}
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} \_2-10 »$ Categories: Kuznetsov's Problem Book Integrals Problem $2 \mid$ Integrals

- Last edited: 19:32, 25 February 2009.
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## Problem Kuznetsov Integrals 2-11

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