Integrate (x+1)sin(x)

\int\left(x+1\right)\sin\left(x\right)dx

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Answer

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

Step by step solution

Problem

$\int\left(x+1\right)\sin\left(x\right)dx$
1

Use the integration by parts theorem to calculate the integral $\int\left(1+x\right)\sin\left(x\right)dx$, using the following formula

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

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=\left(1+x\right)}\\ \displaystyle{du=dx}\end{matrix}$
3

Now, identify $dv$ and calculate $v$

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

Solve the integral

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

Apply the integral of the sine function

$-\cos\left(x\right)$
6

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

$\int\cos\left(x\right)dx-\left(1+x\right)\cos\left(x\right)$
7

Apply the integral of the cosine function

$\sin\left(x\right)-\left(1+x\right)\cos\left(x\right)$
8

Multiply $\left(x+1\right)$ by $-1$

$\sin\left(x\right)+\left(-x-1\right)\cos\left(x\right)$
9

Multiplying polynomials $\cos\left(x\right)$ and $-x+-1$

$\sin\left(x\right)-\cos\left(x\right)-x\cos\left(x\right)$
10

Add the constant of integration

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

Answer

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

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Problem Analysis

Main topic:

Integration by parts

Time to solve it:

0.29 seconds

Views:

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