How do you use partial fraction decomposition to decompose the fraction to integrate (x^3+x^2+x+2)/(x^4+x^2)?

1 Answer
Sep 25, 2015

See the explanation.

Explanation:

x^4+x^2=x^2(x^2+1)

A/x + B/x^2 + (Cx+D)/(x^2+1) =

=(Ax(x^2+1) + B(x^2+1) + (Cx+D)x^2)/(x^2(x^2+1))=

=(Ax^3+Ax+Bx^2+B+Cx^3+Dx^2)/(x^2(x^2+1))=

=((A+C)x^3+(B+D)x^2+Ax+B)/(x^2(x^2+1))

A+C=1
B+D=1
A=1
B=2

C=1-A=0
D=1-B=-1

int (x^3+x^2+x+2)/(x^4+x^2)dx=int dx/x + 2int dx/x^2 - int dx/(x^2+1) =

=ln|x| - 2/x - arctanx + C