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

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Answer 1 · 2015-09-25T16:59:37

See the 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#