# Integral of x^2cos(x)

## \int x^2\cos\left(x\right)dx

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$x^2\sin\left(x\right)-2\sin\left(x\right)+2x\cos\left(x\right)+C_0$

## Step by step solution

Problem

$\int x^2\cos\left(x\right)dx$
1

Use the integration by parts theorem to calculate the integral $\int x^2\cos\left(x\right)dx$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
2

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=x^2}\\ \displaystyle{du=2xdx}\end{matrix}$
3

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\cos\left(x\right)dx}\\ \displaystyle{\int dv=\int \cos\left(x\right)dx}\end{matrix}$
4

Solve the integral

$v=\int\cos\left(x\right)dx$
5

Apply the integral of the cosine function

$\sin\left(x\right)$
6

Now replace the values of $u$, $du$ and $v$ in the last formula

$x^2\sin\left(x\right)-2\int x\sin\left(x\right)dx$
7

Use the integration by parts theorem to calculate the integral $\int x\sin\left(x\right)dx$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
8

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=x}\\ \displaystyle{du=dx}\end{matrix}$
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Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\sin\left(x\right)dx}\\ \displaystyle{\int dv=\int \sin\left(x\right)dx}\end{matrix}$
10

Solve the integral

$v=\int\sin\left(x\right)dx$
11

Apply the integral of the sine function

$x^2\sin\left(x\right)-2\int x\sin\left(x\right)dx$
12

Now replace the values of $u$, $du$ and $v$ in the last formula

$x^2\sin\left(x\right)-2\left(\int\cos\left(x\right)dx-x\cos\left(x\right)\right)$
13

Apply the integral of the cosine function

$x^2\sin\left(x\right)-2\left(\sin\left(x\right)-x\cos\left(x\right)\right)$
14

Multiply $\left(-x\cos\left(x\right)+\sin\left(x\right)\right)$ by $-2$

$x^2\sin\left(x\right)-2\sin\left(x\right)+2x\cos\left(x\right)$
15

$x^2\sin\left(x\right)-2\sin\left(x\right)+2x\cos\left(x\right)+C_0$

$x^2\sin\left(x\right)-2\sin\left(x\right)+2x\cos\left(x\right)+C_0$

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### Main topic:

Integration by parts

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