## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{\pi}\left(2 x^{2}+4 x+7\right) \cos 2 x \, d x
$$
## Solution

$$
\int_{0}^{\pi}\left(2 x^{2}+4 x+7\right) \cos 2 x d x=
$$

Let's denote:

$$
\begin{aligned}
& u=2 x^{2}+4 x+7 ; d u=(4 x+4) d x \\
& d v=\cos 2 x d x ; v=\frac{1}{2} \sin 2 x
\end{aligned}
$$

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

$$
\begin{aligned}
& =\left.\left(2 x^{2}+4 x+7\right) \cdot \frac{1}{2} \sin 2 x\right|_{0} ^{\pi}-\int_{0}^{\pi} \frac{1}{2} \sin 2 x(4 x+4) d x= \\
& =\left(2 \pi^{2}+4 \cdot \pi+7\right) \cdot \frac{1}{2} \sin (2 \cdot \pi)-\left(2 \cdot 0^{2}+4 \cdot 0+7\right) \cdot \frac{1}{2} \sin (2 \cdot 0)-\frac{1}{2} \int_{0}^{\pi} \sin 2 x(2 x+2) d x \\
& =\left(2 \cdot \pi^{2}+4 \pi+7\right) \cdot \frac{1}{2} \cdot 0-7 \cdot \frac{1}{2} \cdot 0-\int_{0}^{\pi} \sin 2 x(2 x+2) d x= \\
& =-\int_{0}^{\pi} \sin 2 x(2 x+2) d x=
\end{aligned}
$$

Let's denote:

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

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

$$
=-\left(\left.(2 x+2) \cdot\left(-\frac{1}{2} \cos 2 x\right)\right|_{0} ^{\pi}-\int_{0}^{\pi}\left(-\frac{1}{2} \cos 2 x\right) \cdot 2 d x\right)=
$$

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

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\%D1\%82 $\% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} \_2-6 \%$ Categories: Kuznetsov's Problem Book Integrals Problem 2 | Integrals

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