(0 to +∞) ∫(dx) / (sqrt(x)(x+1)) , what is the result of this integral?

Answer 1

Let x =u²

Then, dx = 2u du

Then, using I[a,b] for the integral from a to b and E[a,b] for the evaluation from a to b, u also ranges from 0 to ∞

Then, we have I[0, ∞) 2 u du/u (u^2 + 1) du = I[0, ∞) 2 du/(u^2 + 1) du = 2 arctan (u) E[0, ∞) = 2(π/2 = 0) =

π

Answer 2

∫du/sqrt(u² - a²) = ln(u+sqrt(u²-a²)) +C

I didn't do this one myself...I recognized it and looked it up in a table.

To get your problem in this form, complete the square under the radical sign, then make the substitution, y=x+1/2.!