# Help with integration with partial fractions

#### TriForce

So i need to integrate:

$\frac{5x^3+12x^2+12x+10}{(x^2+4)(x^2+2x+1)}$

$=\frac{5x^3+12x^2+12x+10}{(x^2+4)(x+1)^2}$

Partial fractions attempt:

$\frac{Ax+B}{x^2+4}$ + $\frac{C}{x+1}$ + $\frac{D}{(x+1)^2}$

$(Ax+B)(x+1)^2=(Ax+B)(x^2+2x+1)=Ax^3+2Ax^2+Ax+Bx^2+2Bx+B$
$C(x^2+4)(x+1)=C(x^3+x^2+4x+4)=Cx^3+Cx^2+4Cx+4C$
$D(x^2+4)=Dx^2+4D$

$x^3(A+C)=5$

$x^2(2A+B+C+D)=12$

$x^1(A+2B+4C+4D)=12$

$x^0(B+4C+4D)=10$

$A=\frac{32}{5}$
$B=-\frac{22}{5}$
$C=-\frac{7}{5}$
$D=5$

Is this wrong so far? Mathematica seems to think so.

#### SlipEternal

MHF Helper
Try it out.

Wolfram|Alpha: Computational Knowledge Engine)

It is not identically zero, so you made a mistake somewhere.

Let's go through it a bit more carefully.

$\begin{matrix} & Ax^3 & + & (2A+B)x^2 & + & (A+2B)x & + & B\\+ & Cx^3 & + & Cx^2 & + & 4Cx & + & 4C \\ & & + & Dx^2 & & & + & 4D \\ \hline & (A+C)x^3 & + & (2A+B+C+D)x^2 & + & (A+2B+4C)x & + & (B+4C+4D)\end{matrix}$

The difference between this chart and what you wrote is with the $x^1$-term.

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1 person

#### Plato

MHF Helper
So i need to integrate:

$\frac{5x^3+12x^2+12x+10}{(x^2+4)(x^2+2x+1)}$

$=\frac{5x^3+12x^2+12x+10}{(x^2+4)(x+1)^2}$

Partial fractions attempt:

$\frac{Ax+B}{x^2+4}$ + $\frac{C}{x+1}$ + $\frac{D}{(x+1)^2}$

$(Ax+B)(x+1)^2=(Ax+B)(x^2+2x+1)=Ax^3+2Ax^2+Ax+Bx^2+2Bx+B$
$C(x^2+4)(x+1)=C(x^3+x^2+4x+4)=Cx^3+Cx^2+4Cx+4C$
$D(x^2+4)=Dx^2+4D$

$x^3(A+C)=5$

$x^2(2A+B+C+D)=12$

$x^1(A+2B+4C+4D)=12$

$x^0(B+4C+4D)=10$

$A=\frac{32}{5}$
$B=-\frac{22}{5}$
$C=-\frac{7}{5}$
$D=5$

Is this wrong so far? Mathematica seems to think so.
I hate this kind of busywork. I just use what I can. SEE HERE

1 person

#### TriForce

@SlipEternal
This is very helpful. I'll give it another try!