How to integrate by parts

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Sometimes while integrating, we may come across an expression that is not a polynomial, and thus we cannot use the convenient power rule to integrate. Consider the function y=xcos(x). It is not immediately clear how we should start we this one, however upon further inspection, we may introduce the technique of Integrating by Parts. Essentially we split the function into two parts, say u and v, and then employ a formula which allows us to integrate them together:

∫u·dv = u·v − ∫v·du

Applying this to our function, we obtain

∫x·cos(x) dx = u·v − ∫v·du

Here we note that integrating cos(x) is a lot simpler than integrating x, and differentiating x is also simpler than differentiating cos(x), so it would make sense to set

u = x and dv = cos(x)

This in turn gives us

du = 1 and v = sin(x)

Thus plugging these values back into our original formula, we get

∫x·cos(x)= x·sin(x) - ∫sin(x)·1

So now, all we need to do is integrate sin(x), which is definitely easier than what we started with. Thus, the end product gives us 

∫x·cos(x)= x·sin(x) + cos(x) + C

where C of course is the constant of integration.

Dennis L. IB Maths tutor, A Level Maths tutor, GCSE Maths tutor, GCSE...

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