How do I integrate x-1/x+1?

Question

How do I integrate x-1/x+1? I think I need to separate the fraction into

x/x+1 - 1/x+1. but I'm not sure what to do with x/x+1

Thanks

Answer 1

Do polynomial division:

............... 1

........ ______

x + 1 ) x - 1

....... -(x + 1)

....... ————

.............. -2

Thus:

∫ ( x - 1 ) / ( x + 1 ) dx

= ∫ [ 1 - 2/( x + 1 ) ] dx

= ∫ 1 dx - 2·∫ 1/( x + 1 )dx

= x - 2·ln| x + 1 | + C

Answer 2

Instead of splitting it up as you did, you rather want to write:

(x - 1)/(x + 1) = [(x + 1) - 2]/(x + 1) = (x + 1)/(x + 1) - 2/(x + 1) = 1 - 2/(x + 1)

Then, Integral{[(x - 1)/(x + 1)] dx} = Integral{[1 - 2/(x + 1)] dx} = x - 2ln(|x + 1|) + C