Integral of (5+sqrt(x)/5-sqrt(x))dx?

Question

Comment if any extra clarification is necessary.

Answer 1

Hi Martin, This one gets a little tricky, but here are some thoughts.

First let u = sqrt(x). then du = 1/2sqrt(x) dx => dx = 2sqrt(x)du = 2udu.

Substituting all the above gives ∫ (5+u)/(5-u) 2udu, or 2∫(u²+5u)/(5-u)du, which can be rewritten (to make the long division to follow a bit easier): -2 ∫ (u²+5u)/(u-5) du. Use long division to rewrite the integrand:

-2 ∫ (u + 10 + 50/(u-5)) du. Find the anti-derivative of each term:

-2 [ u²/2 + 10 u + 50 ln (u-5) + c ]

Distribute the -2 and substitute sqrt(x) for u. That should do it.