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Step-by-step Solution

Solve the trigonometric integral $\int x^2\cos\left(ax\right)dx$

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Answer

$\frac{-2\left(\frac{-x\cos\left(ax\right)}{a}+\frac{\sin\left(ax\right)}{a^2}\right)+x^2\sin\left(ax\right)}{a}+C_0$

Step-by-step explanation

Problem to solve:

$\int x^2\left(cos\left(ax\right)\right)dx$
1

Use the integration by parts theorem to calculate the integral $\int x^2\cos\left(ax\right)dx$, using the following formula

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

First, identify $u$ and calculate $du$

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

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Answer

$\frac{-2\left(\frac{-x\cos\left(ax\right)}{a}+\frac{\sin\left(ax\right)}{a^2}\right)+x^2\sin\left(ax\right)}{a}+C_0$