please help me with this improper integral question!!!!?

Question

how do you find:

integral from 0 to 1 of: (x+1)/ [sq root((x^2)+2x)] dx


thank you!

Answer 1

1
∫ (x+1)/ [sq root((x^2)+2x)] dx
0

Let u = x^2 + 2x, du = 2x + 2 = 2(x + 1)dx

1
∫ (x+1)/ [sq root((x^2)+2x)] dx =
0
. . . . 3
(1/2) ∫ du/(u)^(1/2) = 2(1/2)u^(1/2) = √3 - 0 = √3
. . . . 0

Answer 2

Just by starring at the problem for a moment, I see that the indefinite integral is sqrt(x^2+2x). Then, substituting in the 0 and 1, you get sqrt(1+2(1))-sqrt(0+2(0)) = sqrt(3)-0 = sqrt(3)