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

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