## Solution

$$
\int_{0}^{2 \pi}\left(3-7 x^{2}\right) \cos 2 x d x=
$$

Let:

$$
\begin{aligned}
& u=3-7 x^{2} ; d u=-14 x \cdot d x \\
& d v=\cos 2 x d x ; v=\frac{1}{2} \sin 2 x
\end{aligned}
$$

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

$$
\begin{aligned}
& =\left.\left(3-7 x^{2}\right) \cdot \frac{1}{2} \sin 2 x\right|_{0} ^{2 \pi}-\int_{0}^{2 \pi} \frac{1}{2} \sin 2 x \cdot(-14 x) \cdot d x= \\
& =\left(3-7 \cdot(2 \pi)^{2}\right) \cdot \frac{1}{2} \sin (2 \cdot 2 \pi)-\left(3-7 \cdot 0^{2}\right) \cdot \frac{1}{2} \sin (2 \cdot 0)+7 \int_{0}^{2 \pi} \sin 2 x \cdot x \cdot d x= \\
& =\left(3-28 \pi^{2}\right) \cdot \frac{1}{2} \cdot 0-3 \cdot \frac{1}{2} \cdot 0+7 \int_{0}^{2 \pi} \sin 2 x \cdot x \cdot d x=7 \int_{0}^{2 \pi} \sin 2 x \cdot x \cdot d x=
\end{aligned}
$$

Let:

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

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

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

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+2-11$ » Categories: Kuznetsov's Problem Book Integrals Problem 2 | Integrals

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## Problem Kuznetsov Integrals 2-12

## Material from PlusPi
## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{2 \pi}\left(3-7 x^{2}\right) \cos 2 x \, d x
$$