## Problem Statement

Calculate the definite integral:

$$
\int_{-3}^{0}\left(x^{2}+6 x+9\right) \sin 2 x \, d x
$$
## Solution

$$
\int_{-3}^{0}\left(x^{2}+6 x+9\right) \sin 2 x d x=
$$

Let:

$$
\begin{aligned}
& u=x^{2}+6 x+9 ; d u=(2 x+6) d x \\
& d v=\sin 2 x d x ; v=-\frac{1}{2} \cos 2 x
\end{aligned}
$$

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

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

Let:

$$
\begin{aligned}
& u=x+3 ; d u=d x \\
& d v=\cos 2 x d x ; v=\frac{1}{2} \sin 2 x
\end{aligned}
$$

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

$$
\begin{aligned}
& =-\frac{9}{2}+\left.(x+3) \cdot \frac{1}{2} \sin 2 x\right|_{-3} ^{0}-\int_{-3}^{0} \frac{1}{2} \sin 2 x d x= \\
& =-\frac{9}{2}+(0+3) \cdot \frac{1}{2} \sin (2 \cdot 0)-(-3+3) \cdot \frac{1}{2} \sin (2 \cdot(-3))-\frac{1}{2} \int_{-3}^{0} \sin 2 x d x= \\
& =-\frac{9}{2}+3 \cdot \frac{1}{2} \cdot 0-0 \cdot \frac{1}{2} \cdot \sin (-6)-\left.\frac{1}{2}\left(-\frac{1}{2} \cos 2 x\right)\right|_{-3} ^{0}= \\
& =-\frac{9}{2}+\frac{1}{4} \cos (2 \cdot 0)-\frac{1}{4} \cos (2 \cdot(-3))=-\frac{18}{4}+\frac{1}{4} \cdot 1-\frac{1}{4} \cos (-6)=-\frac{17+\cos 6}{4}
\end{aligned}
$$

% D1%8B_2-17")

Categories: Kuznetsov Integral Problems Problem 2 | Integrals

- Last modified: 22:11, 26 February 2009.
- Content is available under CC-BY-SA 3.0.

Created by GeeTeatoo

## Problem Kuznetsov Integrals 2-18

## Material from PlusPi