I'm interested in the steps for this problem as well, can someone answer plz
First let y = sqrt(x). Then 2y dy = dx (after a little work.)
So the integral becomes:
Int[sqrt(x)/(x + 100) dx] = Int[2y^2/(y^2 + 100) dy] = 2*Int[y^2/(y^2 + 100) dy]
Now for a trick add and subtract 100 in the numerator:
2*Int[(y^2 + 100 - 100)/(y^2 + 100) dy] = 2*Int[(y^2 + 100)/(y^2 + 100) dy] + 2*Int[-100/(y^2 + 100) dy]
= 2*Int[dy] - 200*Int[1/(y^2 + 100) dy] = 2*y - 20*cot^(-1)(10/y)
= 2*sqrt(x) - 20*cot^(-1)(10/sqrt(x))
-Dan