Solving integrals help!

what is the integral of (2x+5)/(2x-5)dx??

Thanks =)

Hello,

∫ [(2x + 5) /(2x - 5)] dx =

being numerator and denominator of the same order, let's subtract and add 5 in the numerator:

∫ [(2x - 5 + 5 + 5) /(2x - 5)] dx =

∫ {[(2x - 5) + 10] /(2x - 5)} dx =

(distributing and simplifying)

∫ {[(2x - 5) /(2x - 5)] + [10 /(2x - 5)]} dx =

∫ {1 + [10 /(2x - 5)]} dx =

(splitting into two integrals)

∫ dx + ∫ [10 /(2x - 5)] dx =

x + ∫ [10 /(2x - 5)] dx =

let's pull 5 out of the remaining integral, obtaining in the numerator the derivative of the denominator:

x + ∫ [5(2) /(2x - 5)] dx =

x + 5 ∫ [2 /(2x - 5)] dx =

x + 5 ∫ d(2x - 5) /(2x - 5) =

x + 5 ln |2x - 5| + C


the answer is:

∫ [(2x + 5) /(2x - 5)] dx = x + 5 ln |2x - 5| + C


I hope it helps