Integral with three parts?

Question

Say you’re given the integral 5t*cos(2t)e to the t squared how would you go about approaching this? I was thinking about it for a bit and the only viable approach I can see is that you would have to go about using integration by parts. However my question is how given that you have... UV*e? An integral by parts of three pieces?

Answer 1

$e$ is just a constant, so just pass it out of the integral, and then go with your gut feeling. Integration by parts is the way to go!

Edit: the original term is actually $e^{t^2}$ in which case you must use parts like Robert Israel suggests because the derivative includes a $t$ term. However, as per the OP's request, I'm leaving this answer since it includes the integration with $3$ parts.

A worthwhile remark is that the same technique used to integrate by parts will work if for some reason you did have three functions. I must admit I've never needed such a formula in practice, but I'll include it for completeness's sake.

We have the product rule $(uvw)' = u'vw + uv'w + uvw'$. Thus integrating both sides, we obtain the formula:

$$uvw = \int u'vw + \int uv'w + \int uvw'$$

So we can get a formula of the form:

$$\int uvw' = uvw - \int u'vw - \int uv'w $$

It won't treat your example because of the $e^{t^2}$ term not having an integral expressible in elementary functions. However, some terms of it will work out because differentiating will give some $t$ terms, so maybe there's some way to make it work from there. In fact if you do this on your term, things actually get worse, not better, because the natural thing to do, to try to differentiate away the polynomial, actually produces terms that have a larger degree polynomial also. For example, taking $du = \cos (2t)$, $v = 5t$ and $w = e^{t^2}$ (this guy definitely can't be $du$), the last integral is $$\frac{5}{2} \int t^2 \sin(2t) e^{t^2}$$ and that is definitely not any better than where we started from.

Answer 2

I assume you mean $$\int 5 t \cos(2t) e^{t^2}\; dt$$ You might start by noticing that $$5 t e^{t^2} = \frac{5}{2} \dfrac{d}{dt} e^{t^2}$$ and use this in integration by parts. However, you'll still end up with a non-elementary integral. The result can be written as

$$ \frac54\,\sqrt {\pi}{\rm e}\; {\rm erf} \left(it+1\right)-\frac54\,\sqrt {\pi}{ \rm e}\; {\rm erf} \left(it-1\right)+\frac52\,{{\rm e}^{{t}^{2}}}\cos \left( 2\,t \right) $$

EDIT: OK, now you have a completely different integral. As far as I know this has no closed form at all.