Integration in general terms involing $f(x)$ and $f''(x)$

Question

I saw a question asked online and I'm not sure how to approach it. My gut reaction was that it depends on what $f(x)$ actually is. I tried repeated integration by parts but ended up in a mess.

So, is the following expression valid and if so what is the solution? How does one approach something like this?

I've very rarely, if ever, worked with a very general expression like this.

$$ I = \int e^x [f(x) + x(f'(x))]\space dx $$

Answer 1

Using the fact that $f(x) + x f'(x) = \frac{d}{dx}(x\, f(x))$ and integrating by parts, we have: \begin{align} \int e^x\left[f(x) + x f'(x)\right]\, dx &= \int e^x\, \frac{d}{dx}(x\, f(x))\, dx\\ &= e^x\, x\, f(x) \;-\;\int e^x\, x\, f(x)\, dx \end{align} I don't believe this can be simplified further without some knowledge of $f(x)$.

Answer 2

$I = \displaystyle \int e^x\left(xf(x)\right)'dx= \displaystyle \int e^xd\left(xf(x)\right) = e^x(xf(x)) - \displaystyle \int xe^xf(x)dx= xe^xf(x) - \displaystyle \int xe^xf(x)dx.$ Is this helpful enough to meet your requirement?