How do you use partial fraction decomposition to decompose the fraction to integrate #(2x+1)/((x+1)^2(x^2+4)^2)#?

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Answer 1 · 2015-09-26T23:45:54

If the denominator of your rational expression has repeated unfactorable polynomials, then you use linear-factor numerators.

#(2x+1)/((x+1)^2(x^2+4)^2)# = #A/(x+1)#+#B/(x+1)^2#+#(Cx+D)/(x^2+4)#+#(Ex+F)/(x^2+4)^2#

Next, multiply both sides of the equation by #(x+1)^2(x^2+4)^2#

#2x+1# = #A(x+1)(x^2+4)^2#+#B(x^2+4)^2#+#(Cx+D)(x+1)^2(x^2+4)#+#(Ex+F)(x+1)^2#

Next, expand each expression on the right and match up to the values on left side of the equation. Solve for the unknown constants A - F.

You should get the following solutions:

#(13-8 x)/(25 (x^2+4)^2)+(11-6 x)/(125 (x^2+4))+6/(125 (x+1))-1/(25 (x+1)^2)#

Hope that helped