<s>[INST] Compute $\int x^2 \cos \frac{x}{2} \mathrm{d}x$ I am trying to compute the following integral:
$$\int x^2 \cos \frac{x}{2} \mathrm{d}x$$
I know this requires integration by parts multiple times but I am having trouble figuring out what to do once you have integrated twice. This is what I have done:
Let $u = \cos \frac{x}{2}$ and $\mathrm{d}u = \frac{-\sin\left(\dfrac{x}{2}\right)}{2} \mathrm{d}x$ and $\mathrm{d}v = x^2$ and $v= \frac{x^3}{3}$. 
\begin{align}
&\int x^2 \cos \frac{x}{2} \\
&\cos \frac{x}{2} \cdot \frac{x^3}{3} - \int \frac{x^3}{3} \cdot \frac{-\sin\left(\dfrac{x}{2}\right)}{2} \mathrm{d}x
\end{align}
So now I integrate $\int \frac{x^3}{3} \cdot \frac{-\sin\left(\dfrac{x}{2}\right)}{2} \mathrm{d}x$ to get: 
\begin{align}
\frac{-\sin\left(\dfrac{x}{2}\right)}{2} \cdot \frac{x^4}{12} - \int \frac{x^4}{12} \cdot \frac{\cos x}{2}
\end{align}
Now, this is where I get stuck. I know if I continue, I will end up with $\frac{-\sin\left(\dfrac{x}{2}\right)}{2}$ again when I integrate $\cos \frac{x}{2}$. So, where do I go from here? 
Thanks!
 [/INST] Hint: Switch the functions you originally set equal to $u$ and $dv$.
General Strategy: Assume you're trying to compute $\int p(x) f(x) dx$ where $p(x) = a_0 + a_1 x + \ldots + a_nx^n$ is a polynomial and $f(x)$ is a function that you know how to integrate $n$ times. (For example, in your problem $p(x) = x^2$, $f(x) = \cos x$, and we note that it's easy to integrate $\cos x$ twice.) Then you can just keep integrating by parts ($n$ times), setting $u = p(x)$ (or $p^{(k)}(x)$) and $dv = \lbrack\text{whatever is leftover}\rbrack dx$ to eventually arrive at
$$
\int p(x) f(x) dx = \lbrack\text{a bunch of terms not involving integrals}\rbrack + \int F(x) dx
$$
where $F(x)$ is a function such that $F^{(n)}(x) = f(x)$.
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