Determine if the following definite integrals is convergent, and if it is, give its value:-?

Question

i) ∫ x e^(-x^2) ; from 0 to infinity

ii) ∫ dx/(x - 2 + x^2) ; from 0 to 1

Answer 1

The key to doing these sort of integrals is to do the integration to a defined value and then let it approach the actual one you want.

So in Q1 you integrate from 0 to n and then let n become very large and see whether the result has a limit. This integral is pretty easy. Try differentiating e^(-x^2).

For Q2 integrate from 0 to n and then let n approach 1 and see whether the result has a limit. To do this integral note that

x^2 + x - 2 = (x + 1/2)^2 - 9/4

and use the substitution

x + 1/2 = (3/2)secA

together with the identity

(tanA)^2 = (secA)^2 - 1

Don't forget that dx = (3/2)secA*tanA dA

Try to finish them yourself before looking at any more detailed answer left here.