# Step-by-step Solution

## Trigonometric integral $\int x^3\sin\left(x\right)dx$

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### Videos

$-x^3\cos\left(x\right)+3x^{2}\sin\left(x\right)-6\left(-x\cos\left(x\right)+\sin\left(x\right)\right)+C_0$

## Step-by-step explanation

Problem to solve:

$\int\:x^3sen\:\left(x\right)dx$
1

Use the integration by parts theorem to calculate the integral $\int x^3\sin\left(x\right)dx$, using the following formula

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

First, identify $u$ and calculate $du$

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

$-x^3\cos\left(x\right)+3x^{2}\sin\left(x\right)-6\left(-x\cos\left(x\right)+\sin\left(x\right)\right)+C_0$
$\int\:x^3sen\:\left(x\right)dx$

### Main topic:

Integration by parts

~ 0.85 seconds

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