Step-by-step Solution

Integrate $x^2\sin\left(x\right)$ with respect to x

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Final Answer

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

Step-by-step explanation

Problem to solve:

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

Choose the solving method

1

We can solve the integral $\int x^2\sin\left(x\right)dx$ by applying the method of tabular integration by parts, which allows us to perform successive integrations by parts on integrals of the form $\int P(x)T(x) dx$. $P(x)$ is typically a polynomial function and $T(x)$ is a transcendent function such as $\sin(x)$, $\cos(x)$ and $e^x$. The first step is to choose functions $P(x)$ and $T(x)$

$\begin{matrix}P(x)=x^2 \\ T(x)=\sin\left(x\right)\end{matrix}$
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Derive $P(x)$ until it becomes $0$

$0$
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Integrate $T(x)$ as many times as we have had to derive $P(x)$

$\cos\left(x\right)$
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With the derivatives and integrals of both functions we build the following table

$\begin{matrix}\mathrm{Derivatives} & \mathrm{Sign} & \mathrm{Integrals} \\ & & \sin\left(x\right) \\ x^2 & + & -\cos\left(x\right) \\ 2x & - & -\sin\left(x\right) \\ 2 & + & \cos\left(x\right) \\ 0 & & \end{matrix}$
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Then the solution is the sum of the products of the derivatives and the integrals according to the previous table. The first term consists of the product of the polynomial function by the first integral. The second term is the product of the first derivative by the second integral, and so on.

$-x^2\cos\left(x\right)+2x\sin\left(x\right)+2\cos\left(x\right)$
6

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

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

Main topic:

Calculus

Related formulas:

4. See formulas

Time to solve it:

~ 0.05 s (SnapXam)