How do you integrate (x^3+2)/(x^2-x) using partial fractions?

1 Answer
Narad T.
Oct 18, 2016

=x^2/2+x-2lnx+3ln(x-1)+C

Explanation:

First make a long division to determine
(x^3+2)/(x^2-x)=x+1+(x+2)/(x^2-x)
To simplify the fraction we use partial fraction
(x+2)/(x^2-x)=(x+2)/(x(x-1))=A/x+B/(x-1)=(A(x-1)+Bx)/(x(x-1))
So we determine A and B
x+2=A(x-1)+Bx
Cmparing the coefficients, we find
1=A+B and 2=-A and we conclude B=3
int((x^3+2)dx)/(x^2-x)=int(x+1)dx-int2dx/x+int3dx/(x-1)
=x^2/2+x-2lnx+3ln(x-1)+C

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