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Homework Statement
Evaluate ∫-x/(x+1) dx
Homework Equations
The Attempt at a Solution
I do a U-sub
u=x+1
du=dx
x=u-1
-∫(u-1)/u du
-∫[u/u - 1/u] du
-∫1 - 1/u du
-[∫1du - ∫1\u du]
-[u - ln(u) + C]
ln(u)-u+C
ln(x+1)-(x+1)+C
ln(x+1)-x-1+C
I combine -1 and C into a single C to get the final answer
ln(x+1) - x + CWolfram says the answer is correct, but its method is slightly different. It uses long division at the outset to go from -∫x/(x+1) dx → -∫1 - 1/(x+1) dx before seperating into 2 integrals, and doing a u-sub on the second. My answer ends up the same, but getting that extra -1 from substituting the (x+1) in for u is making me nervous for some reason, even though I ultimately combine it with the constant. Is my process equally correct (I think it is) or did I just get lucky with the solution?
Thanks
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