We can solve the integral $\int\left(x^2+x\right)\cos\left(x\right)dx$ 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 $T(x)$

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.

Multiply the single term $\sin\left(x\right)$ by each term of the polynomial $\left(x^2+x\right)$

Solve the product $\left(2x+1\right)\cos\left(x\right)$

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