MHB Partial Fraction Decomposition Help - Calculus BC

MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Here is the question:

Help with Calculus BC: partial fractions!?

Thank you for your help guys. I wrote the problem wrong in the previous question :(

integral (4x^2+x-2)/(x^3+2x^2-3x) dx

howwwww?

Here is a link to the question:

Help with Calculus BC: partial fractions!? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
Mathematics news on Phys.org
Hello living higher:

We are given to decompose:

$\displaystyle \frac{4x^2+x-2}{x^3+2x^2-3x}$

The first step is to factorize the denominator:

$x^3+2x^2-3x=x(x^2+2x-3)=x(x-1)(x+3)$

Now we assume the decomposition will take the form:

$\displaystyle \frac{4x^2+x-2}{x(x-1)(x+3)}=\frac{A}{x+3}+\frac{B}{x}+\frac{C}{x-1}$

Now, rather than set up a linear system of equations, I am going to use a shortcut method called the Heaviside cover-up method. Look at the first term on the right. The root of the denominator is $x=-3$.

To find the value of $A$, we "cover-up" the factor $x+3$ to get:

$\displaystyle \frac{4x^2+x-2}{x(x-1)}$

and we evaluate this at $x=-3$ to find:

$\displaystyle A=\frac{4(-3)^2+(-3)-2}{(-3)((-3)-1)}=\frac{31}{12}$

Next we look at the second term in the decomposition and we see the root of the denominator is $x=0$, and covering up the factor $x$ on the left, and evaluating it for $x=0$, we find:

$\displaystyle B=\frac{4(0)^2+(0)-2}{((0)-1)((0)+3)}=\frac{2}{3}$

And finally, we look at the third term in the decomposition and we see the root of the denominator is $x=1$, and covering the the factor $x-1$ on the left and evaluation it for $x=1$, we find:

$\displaystyle C=\frac{4(1)^2+(1)-2}{(1)((1)+3)}=\frac{3}{4}$

And now we may state:

$\displaystyle \frac{4x^2+x-2}{x^3+2x^2-3x}=\frac{31}{12(x+3)}+\frac{2}{3x}+\frac{3}{4(x-1)}$

Now we may directly integrate:

$\displaystyle \int\frac{31}{12(x+3)}+\frac{2}{3x}+\frac{3}{4(x-1)}\,dx=\frac{31}{12}\ln|x+3|+\frac{2}{3}\ln|x|+ \frac{3}{4}\ln|x-1|+C$
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top