How do you choose which substitutions to make when integrating by substitution?
We make substitutions in order to make integrations simpler. It is therefore useful to make substitutions that bring together terms inside functions. For example: when integrating f(x)= cos(3x -1) it would be useful to make the substitution u = 3x-1 making the integration cos(u).Another thing to look for is whether a substitution can be differentiated to be equal to a term in the integrand, as when you change the integration measure it will eliminate that term. For example in the function f(x)= cos(x)sin2(x) it would be useful to make the substitution u = sin(x). When this is differentiated to find du it gives du = cos(x)dx. This eliminates the cosine term leaving an integral over u2.
