Audrina Jackson

2022-07-14

Solving $\int \left(\frac{5{x}^{2}+3x-2}{{x}^{3}+2{x}^{2}}\right)$ via partial fractions
So, first of all, we must factorize the denominator:
${x}^{3}+2{x}^{2}=\left(x+2\right)\cdot {x}^{2}$
Great. So now we write three fractions:
$\frac{A}{{x}^{2}}+\frac{B}{x}+\frac{C}{x+2}$
Eventually we conclude that
$A\left(x+2\right)+B\left(x+2\right)\left(x\right)+C\left({x}^{2}\right)=5{x}^{2}+3x-2$
So now we look at what happens when x=−2:
C=12
When x=0:
A=−1And now we are missing B, but we can just pick an arbitrary number for x like... 1:
B=−1

Zachery Conway

You have an algebra error. When you put x=−2, the constraint equation collapses to 4C=12, so C=3.
Your method for determining B is correct – the constraining identity must hold for any x. Putting x=1 is a fine choice as the arithmetic is easier. However the incorrect value for C will affect the value you determine for B.

Jamison Rios

I think this is the easiest way.
$\left(B+C\right){x}^{2}+\left(A+2B\right)x+2A=5{x}^{2}+3x-2$
$\begin{array}{rlrl}2A& =-2& \to A& =-1\\ A+2B& =-1+2B=3& \to B& =2\\ B+C& =2+C=5& \to C& =3\end{array}$

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