Question

Show that the following integral is convergent,

Integral, it goes from 0 to infinity dx/(sqrt(x) + x^2)

Answer 1

Let u^2 = x
2u du = dx
and now the integral becomes
∫ 2u/(u+u^4) du = 2∫du/(1+u^3) = 2∫du/((u+1)(u^2-u+1)
= 2/3 ∫(1/(u+1) - (u-2)/(u^2-u+1)) du
= 2/3 ∫ 1/(u+1) - (1/2)(2u-1)/(u^2-u+1) + (3/2)/(u^2-u+1) du
= 2/3 ∫ 1/(u+1) - (1/2)(2u-1)/(u^2-u+1) + (3/2)/((u-1/2)^2 + 3/4) du

Now you can see how to do each of those, using log and arctan ... right?
check my math on all those fractions