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Exercise

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

Step-by-step Solution

1

We can solve the integral $\int x^2\sin\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

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

First, identify or choose $u$ and calculate it's derivative, $du$

$\begin{matrix}\displaystyle{u=x^2}\\ \displaystyle{du=2xdx}\end{matrix}$
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 to find $v$

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

Apply the integral of the sine function: $\int\sin(x)dx=-\cos(x)$

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

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

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

We can solve the integral $\int x\cos\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

Make diagrams like this with our diagram generator
$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
8

First, identify or choose $u$ and calculate it's derivative, $du$

$\begin{matrix}\displaystyle{u=x}\\ \displaystyle{du=dx}\end{matrix}$
9

Now, identify $dv$ and calculate $v$

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

Solve the integral to find $v$

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

Apply the integral of the cosine function: $\int\cos(x)dx=\sin(x)$

$\sin\left(x\right)$
12

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

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

Multiply the single term $2$ by each term of the polynomial $\left(x\sin\left(x\right)-\int\sin\left(x\right)dx\right)$

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

The integral $-2\int\sin\left(x\right)dx$ results in: $2\cos\left(x\right)$

$2\cos\left(x\right)$
15

Gather the results of all integrals

$-x^2\cos\left(x\right)+2x\sin\left(x\right)+2\cos\left(x\right)$
16

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

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

Final answer to the exercise

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

Try other ways to solve this exercise

  • Integrate by parts
  • Integrate by partial fractions
  • Integrate by substitution
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Integrate using basic integrals
  • Product of Binomials with Common Term
  • FOIL Method
  • Load more...
Can't find a method? Tell us so we can add it.

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Main Topic: Integration by Parts

Integration by parts is an integration method that relates the integral of a product of two or more functions to the integral of their derivative and antiderivative.

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