MHB Integrating $\cos(mx)$ with Two Variables

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The integration of the function $$\int x^2\cos(mx)\,dx$$ is approached using integration by parts, where $u = x^2$ and $dv = \cos(mx)\,dx$. In this context, $m$ is treated as a constant unless specified otherwise, allowing for straightforward integration with respect to $x$. The integration by parts will yield a solution involving both $x$ and the cosine function. This method effectively addresses the integration of a product of a polynomial and a trigonometric function.
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$$\int x^2\cos(mx)\,dx$$

When integrating this by parts, the $x^2$ will become the $u$ and the $\cos(mx)\,dx$ will become $dv$.

How is the $\cos(mx)$ integrated if there are two variables?
 
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You are integrating with respect to $x$, and unless $m$ is a function of $x$, then you treat it as a constant. :D
 
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