## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{\pi}\left(9 x^{2}+9 x+11\right) \cos 3 x d x
$$
## Solution

$$
\int_{0}^{\pi}\left(9 x^{2}+9 x+11\right) \cos 3 x d x=
$$

Let:

$$
\begin{aligned}
& u=9 x^{2}+9 x+11 ; d u=(18 x+9) d x \\
& d v=\cos 3 x d x ; v=\frac{1}{3} \sin 3 x
\end{aligned}
$$

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

$$
\begin{aligned}
& =\left.\left(9 x^{2}+9 x+11\right) \cdot \frac{1}{3} \sin 3 x\right|_{0} ^{\pi}-\int_{0}^{\pi} \frac{1}{3} \sin 3 x \cdot(18 x+9) \cdot d x= \\
& =\left(\pi^{2}+9 \cdot \pi+11\right) \cdot \frac{1}{3} \sin (3 \cdot \pi)-\left(0^{2}+9 \cdot 0+11\right) \cdot \frac{1}{3} \sin (3 \cdot 0)-\int_{0}^{\pi} \sin 3 x \cdot(6 x+3) \cdot d x= \\
& =\left(\pi^{2}+9 \pi+11\right) \cdot \frac{1}{3} \cdot 0-11 \cdot \frac{1}{3} \cdot 0-\int_{0}^{\pi} \sin 3 x \cdot(6 x+3) \cdot d x=-\int_{0}^{\pi} \sin 3 x \cdot(6 x+3) \cdot d x=
\end{aligned}
$$

Let:

$$
\begin{aligned}
& u=6 x+3 ; d u=6 d x \\
& d v=\sin 3 x d x ; v=-\frac{1}{3} \cos 3 x
\end{aligned}
$$

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

$$
\begin{aligned}
& =-\left(\left.(6 x+3) \cdot\left(-\frac{1}{3} \cos 3 x\right)\right|_{0} ^{\pi}-\int_{0}^{\pi}\left(-\frac{1}{3} \cos 3 x\right) \cdot 6 d x\right)= \\
& =-\left((6 \cdot \pi+3) \cdot\left(-\frac{1}{3} \cos (3 \cdot \pi)\right)-(6 \cdot 0+3) \cdot\left(-\frac{1}{3} \cos (3 \cdot 0)\right)+2 \cdot \int_{0}^{\pi} \cos 3 x d x\right)= \\
& =-\left(2 \pi+1+1+\left.2 \cdot \frac{1}{3} \sin 3 x\right|_{0} ^{\pi}\right)=-\left(2 \pi+2+\frac{2}{3} \sin (3 \cdot \pi)-\frac{2}{3} \sin (3 \cdot 0)\right)=
\end{aligned}
$$

$$
=-(2 \pi+2+0-0)=-2 \pi-2
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\�_\�\� \%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD\%D1\%82\%D0\%B5 $\% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} \_2-7$ >

Categories: Kuznetsov's Problem Book Integrals Problem $2 \mid$ Integrals

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## Problem Kuznetsov Integrals 2-8

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