Integration by parts

Saw this referred to in A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity and it made me shudder – as I always seemed to struggle with it at ‘A’ level maths – so here, for my own benefit, is a quick proof/explanation of the method.

The general rule is:

\int v(x) \frac{du}{dx} dx = u(x)v(x) - \int u(x) \frac{dv}{dx} dx

And why is this so?

From the product rule we know:

\frac{d}{dx} (u(x)v(x)) = \frac{du}{dx} v(x) + \frac{dv}{dx}u(x)

So \frac{du}{dx} v(x) = \frac{d}{dx}(u(x)v(x)) - \frac{dv}{dx}u(x)

And, integrate both sides and we have:

\int v(x) \frac{du}{dx} dx = u(x)v(x) - \int u(x) \frac{dv}{dx} dx

Of course, we still have to apply this sensibly to make our problem easier to integrate!


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