Thread: Trig substitutions for integrals

Hi,

Does anybody know good substitutions to use for integrating:

x^2/(4-x^2)^1/2

and

((4-x^2)^1/2)/x^2

I've been thinking I should use u = sin^-1(x/2) for the first and u = cos^-1(x/2) for the second, but these do not seem to yield happy integrals. The first yields integral of 4sin^2(u) du (which is doable but not especially neat using a cos(2u) substitution), and the second yields integral of -tan^2(u) du, which I do not know how to do (but maybe I should?). Are there better substitutions than these?

Cheers

Originally Posted by DangerousDave

Hi,

Does anybody know good substitutions to use for integrating:

x^2/(4-x^2)^1/2

and

((4-x^2)^1/2)/x^2

I've been thinking I should use u = sin^-1(x/2) for the first and u = cos^-1(x/2) for the second, but these do not seem to yield happy integrals. The first yields integral of 4sin^2(u) du (which is doable but not especially neat using a cos(2u) substitution), and the second yields integral of -tan^2(u) du, which I do not know how to do (but maybe I should?). Are there better substitutions than these?

Your substitutions are spot on, couldn't be better. For the second one, notice that , and the integral of is .