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

Sep 26, 2015

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

#### Explanation:

$\frac{2 x + 1}{{\left(x + 1\right)}^{2} {\left({x}^{2} + 4\right)}^{2}}$ = $\frac{A}{x + 1}$+$\frac{B}{x + 1} ^ 2$+$\frac{C x + D}{{x}^{2} + 4}$+$\frac{E x + F}{{x}^{2} + 4} ^ 2$

Next, multiply both sides of the equation by ${\left(x + 1\right)}^{2} {\left({x}^{2} + 4\right)}^{2}$

$2 x + 1$ = $A \left(x + 1\right) {\left({x}^{2} + 4\right)}^{2}$+$B {\left({x}^{2} + 4\right)}^{2}$+$\left(C x + D\right) {\left(x + 1\right)}^{2} \left({x}^{2} + 4\right)$+$\left(E x + F\right) {\left(x + 1\right)}^{2}$

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:

$\frac{13 - 8 x}{25 {\left({x}^{2} + 4\right)}^{2}} + \frac{11 - 6 x}{125 \left({x}^{2} + 4\right)} + \frac{6}{125 \left(x + 1\right)} - \frac{1}{25 {\left(x + 1\right)}^{2}}$

Hope that helped