$\int f'g'$ given $f$ and $g$

Question

Suppose I know $f(x)$ and $g(x)$. Are there any tricks or techniques for finding $\int f'(x)g'(x) dx$? Or other possible forms of $\int f'(x)g'(x) dx$ that might be useful?

(The title contains the meat of the question.)

Answer 1

When $f(x) = g(x) = e^{x^2}$, $f'(x) = g'(x) = 2xe^{x^2}$.

However, $\displaystyle \int f'(x) g'(x) \ \mathrm dx = \int 4x^2 e^{2x^2} \ \mathrm dx = x e^{2x^2} - \frac{\sqrt{2\pi}}{4} \operatorname{erfi}(\sqrt 2x)$ is non-elementary (thanks to WolframAlpha).

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We have the product rule of differentiation: $(fg)' = f'g + fg'$.

If we integrate both sides, we get $fg = \displaystyle \int (f'g + fg') \ \mathrm dx$.

Rearranging terms gives us $\displaystyle \int fg' \ \mathrm dx = fg - \int f'g \ \mathrm dx$.

Absorbing derivatives gives $\displaystyle \int f \ \mathrm dg = fg - \int g \ \mathrm df$.

This is the integration by part.

So, you can't really convert the product rule to an analog in integration.