## Final Answer

## Step-by-step Solution

Problem to solve:

Choose the solving method

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 $T(x)$

Find the derivative of $x^2$ with respect to $x$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

The derivative of the linear function times a constant, is equal to the constant

The derivative of the constant function ($2$) is equal to zero

Derive $P(x)$ until it becomes $0$

Find the integral of $\sin\left(x\right)$ with respect to $x$

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

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

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

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

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

Integrate $T(x)$ as many times as we have had to derive $P(x)$, so we must integrate $\sin\left(x\right)$ a total of $3$ times

With the derivatives and integrals of both functions we build the following table

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.

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