Determine whether the improper integral is convergent or divergent. If it is convergent, evaluate it.?

Question

5
∫ x^-1/2 + (5-x)^-1/2 dx
0

Answer 1

∫ x^{-1/2} dx = x^{1/2} /(1/2) + C = 2x^{1/2} +C
and
∫ (5-x)^{-1/2} dx = - 2 (5-x)^{1/2} + C
so:
5
∫ x^-1/2 + (5-x)^-1/2 dx
0
= lim p->0 ∫ from p to 5 of x^-1/2 + (5-x)^-1/2 dx
= lim p->0 2(5^{1/2} -p^{1/2}) -2 ( 0^{1/2} - (5-p)^{1/2})
=2 sqrt(5) + 2 sqrt(5) = 4sqrt(5)
i.e. the integral converges

Answer 2

Huge hint: It's "equal" to twice the integral of either one of the two terms. I.e., they both integrate to the same value (if they converge) or else both diverge.