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As you will be aware, it is not easy to directly integrate terms involving sin^2(x) and cos^2(x), so we use a substitution to turn them into something which we can integrate.

From the double a...

The integration by parts formula takes the form:

int(uv') = uv - int(vu') 

where v' = dv/dx and u' = du/dx

A lot of the art of using the integration by part...