# Integrate xsin(x) from pi/2 to ((5*pi)/2)^2

## \int_{\frac{\pi }{2}}^{\left(\frac{5\pi }{2}\right)^2} x\cdot\sin\left(x\right)dx

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$-x\cos\left(x\right)-\sqrt[3]{13}$

## Step by step solution

Problem

$\int_{\frac{\pi }{2}}^{\left(\frac{5\pi }{2}\right)^2} x\cdot\sin\left(x\right)dx$
1

Multiply $5$ times $\pi$

$\int_{\sqrt{2}}^{\left(\frac{15.708}{2}\right)^2} x\sin\left(x\right)dx$
2

Divide $15.708$ by $2$

$\int_{\sqrt{2}}^{7.854^2} x\sin\left(x\right)dx$
3

Calculate the power

$\int_{\sqrt{2}}^{61.685} x\sin\left(x\right)dx$
4

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

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
5

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=x}\\ \displaystyle{du=dx}\end{matrix}$
6

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\sin\left(x\right)dx}\\ \displaystyle{\int dv=\int \sin\left(x\right)dx}\end{matrix}$
7

Solve the integral

$v=\int\sin\left(x\right)dx$
8

Apply the integral of the sine function

$\int_{\sqrt{2}}^{61.685} x\sin\left(x\right)dx$
9

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

$-x\cos\left(x\right)--\int_{\sqrt{2}}^{61.685}\cos\left(x\right)dx$
10

Apply the integral of the cosine function

$-x\cos\left(x\right)-\left[-\sin\left(x\right)\right]_{\sqrt{2}}^{61.685}$
11

Evaluate the definite integral

$\left(\sin\left(61.685\right)\left(-1\right)-1\cdot \sin\left(\sqrt{2}\right)\left(-1\right)\right)\left(-1\right)-x\cos\left(x\right)$
12

Multiply $-1$ times $-1$

$\left(\sin\left(\sqrt{2}\right)\cdot 1+\sin\left(61.685\right)\left(-1\right)\right)\left(-1\right)-x\cos\left(x\right)$
13

Calculating the sine of $61.685$ degrees

$\left(1\cdot 1-\frac{\sqrt{3}}{2}\left(-1\right)\right)\left(-1\right)-x\cos\left(x\right)$
14

Multiply $-1$ times $-\frac{\sqrt{3}}{2}$

$\left(1+\frac{\sqrt{3}}{2}\right)\left(-1\right)-x\cos\left(x\right)$
15

Add the values $\frac{\sqrt{3}}{2}$ and $1$

$\sqrt[3]{13}\left(-1\right)-x\cos\left(x\right)$
16

Multiply $-1$ times $\sqrt[3]{13}$

$-x\cos\left(x\right)-\sqrt[3]{13}$

$-x\cos\left(x\right)-\sqrt[3]{13}$

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### Main topic:

Integration by parts

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