## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{\pi}\left(8 x^{2}+16 x+17\right) \cos 4 x \, dx
$$
## Solution

$$
\int_{0}^{\pi}\left(8 x^{2}+16 x+17\right) \cos 4 x d x=
$$

Let's denote:

$$
\begin{aligned}
& u=8 x^{2}+16 x+17 ; d u=(16 x+16) d x \\
& d v=\cos 4 x d x ; v=\frac{1}{4} \sin 4 x
\end{aligned}
$$

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

$$
\begin{aligned}
& =\left.\left(8 x^{2}+16 x+17\right) \cdot \frac{1}{4} \sin 4 x\right|_{0} ^{\pi}-\int_{0}^{\pi} \frac{1}{4} \sin 4 x \cdot(16 x+16) \cdot d x= \\
& =\left(8 \pi^{2}+16 \cdot \pi+17\right) \cdot \frac{1}{4} \sin (4 \cdot \pi)-\left(8 \cdot 0^{2}+16 \cdot 0+17\right) \cdot \frac{1}{4} \sin (4 \cdot 0)-\int_{0}^{\pi} \sin 4 x \cdot(4 x+ \\
& =\left(8 \pi^{2}+16 \pi+11\right) \cdot \frac{1}{4} \cdot 0-17 \cdot \frac{1}{4} \cdot 0-\int_{0}^{\pi} \sin 4 x \cdot(4 x+4) \cdot d x=-\int_{0}^{\pi} \sin 4 x \cdot(4 x+4) \cdot d x=
\end{aligned}
$$

Let's denote:

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

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

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

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

Source — "http://pluspi.org/wiki/index.php/%D0%97%D0%B0%D0%B4%D0%B0%D1%87%D0%B0_%D0%9A%D1%83%D0%B7%D0%BD%D0%B5%D1%86%D0%BE%D0%B2%D0%B0_%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D1%8B_2-8" Categories: Kuznetsov's Problem Book Integrals Problem 2 | Integrals

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## Problem Kuznetsov Integrals 2-9

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