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Integration by parts Calculator

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1

Solved example of integration by parts

$\int x\cdot\cos\left(x\right)dx$
2

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

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

The derivative of the linear function is equal to $1$

$1$
3

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=x}\\ \displaystyle{du=dx}\end{matrix}$
4

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\cos\left(x\right)dx}\\ \displaystyle{\int dv=\int \cos\left(x\right)dx}\end{matrix}$
5

Solve the integral

$v=\int\cos\left(x\right)dx$
6

Apply the integral of the cosine function

$\sin\left(x\right)$
7

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

$x\sin\left(x\right)-\int\sin\left(x\right)dx$

Apply the integral of the sine function

$\cos\left(x\right)$
8

The integral $-\int\sin\left(x\right)dx$ results in: $\cos\left(x\right)$

$\cos\left(x\right)$
9

After gathering the results of all integrals, the final answer is

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

As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration

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

Answer

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

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