What is Partial fractions: Definition and 297 Discussions
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.The importance of the partial fraction decomposition lies in the fact that it provides algorithms for various computations with rational functions, including the explicit computation of antiderivatives, Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms. The concept was discovered independently in 1702 by both Johann Bernoulli and Gottfried Leibniz.In symbols, the partial fraction decomposition of a rational fraction of the form
f
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x
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g
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x
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,
{\displaystyle \textstyle {\frac {f(x)}{g(x)}},}
where f and g are polynomials, is its expression as
f
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g
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x
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=
p
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x
)
+
∑
j
f
j
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x
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g
j
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x
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{\displaystyle {\frac {f(x)}{g(x)}}=p(x)+\sum _{j}{\frac {f_{j}(x)}{g_{j}(x)}}}
where
p(x) is a polynomial, and, for each j,
the denominator gj (x) is a power of an irreducible polynomial (that is not factorable into polynomials of positive degrees), and
the numerator fj (x) is a polynomial of a smaller degree than the degree of this irreducible polynomial.
When explicit computation is involved, a coarser decomposition is often preferred, which consists of replacing "irreducible polynomial" by "square-free polynomial" in the description of the outcome. This allows replacing polynomial factorization by the much easier to compute square-free factorization. This is sufficient for most applications, and avoids introducing irrational coefficients when the coefficients of the input polynomials are integers or rational numbers.
Homework Statement
dx/dt=3x(x-5); x(0)=2
Homework Equations
dx/dt=F(x)G(t)
dx/dt=x'
The Attempt at a Solution
First I separated the variables using F=3x(x-5) and G=3
x'/(3x(x-5))=3
Applying method of quadrature:
\int\frac{x'}{(3x(x-5)}dt=\int3 dt
On the right hand...
We were discussing them in my math methods class today however I'm not really sure how the idea works.
Does anyone know of any online references that might be of some help? Google wasn't much help for me =/
Homework Statement
It's hard to explain, I can do everything exept get the answer to what I'v pointed out below.
I just don't know what order to solve it into get the correct answer, I must have tried every method except the right one!
Homework Equations
The Attempt at a...
Use partial fractions to integrate x^3/(x^3+1)The Attempt at a Solution
\int x^{3}/(x^{3}+1) dx
Homework Statement
Homework Equations
The Attempt at a Solution
\int x^{3}/x^{3}+1 dx
I know that first i have to perform long division but i am at a loss how to do this
THanks
Homework Statement
\int\frac{e^xcos(log_7(e^x+9))}{(e^x+9)ln(7)}dx
Homework Equations
The Attempt at a Solution
Let u= (ex+9)
du= exdx
New integral \int\frac{cos(log_7(u))}{(u)ln(7)}du
This is where I got l little lost. Should I let log7(u)=\frac{ln(u)}{ln(7)}? Or is this...
Suppose that q(z) = 1, and p(z) = (1 + z)(1 + 3z).
We wish to express q(z)/p(z) in the form
where A and B are constants. To find them, we multiply through by p(z) =
(1 + z)(1 + 3z) and obtain
1 = A(1 + 3z) + B(1 + z)
= (A + B) + (3A + B)z
Im fine up to this point, But according to...
Homework Statement
\int(3x3-4x2-3x+2)/(x4-x2)
Homework Equations
P(x)/Q(x)=A1/(x-r1)+A2/(x-r2)+...
if x-r occurs with multiplicity m, then A/(x-r) must be replaced by a sum of the form:
B1/(x-r)+B2/(x-r)2+...
I think this second equation is the source of my confusion.
The Attempt at a...
1.\int2x^2+x+9/(9x+1)(x^2+9) dx
2. (A/9x+1) + [(Bx + C ) / (x^2 + 9)]
I get the worst numbers when I solve the system. The question is from an old exam and calculators are not allowed. Am I doing something wrong or is there another way to integrate this?
Hello!
Quick question reagrding partial fractions.
When there is a factor such as (x+2)3 in the denominator, then the fraction is separated into the components (x+2)1+...+(x+2)3.
I am not convinced I understand quite why this is so. Partial fractions guides all offer something on the theme...
Homework Statement
The question asks me to express the integrand in partial fractions to evaluate the integral
\int \frac{13x-4}{6x^{2} -x -2} dx
Homework Equations
The Attempt at a Solution
Well 6x² -x - 2 doesn't factorise (or I can't see it factorised).
So I tried...
Homework Statement
Hi,
\int \frac{1}{x(x^{2}+1)}dx
Homework Equations
The Attempt at a Solution
well I split this into partial fractions
\frac{A}{x} + \frac{Bx + C}{x^{2} + 1}
so 1 \equiv A(x^{2}+1) + (Bx + C)x
when x = 0, A =1
when x = 1, Bx + C = -1 so...
Hello,
I'm don't understand a step in the following integral:
∫(x-1)/(2x+1)dx = ∫(1/2)dx − (3/2)∫1/(2x+1)dx = (1/2)x − (3/4)ln|2x+1| + C
The first step, where you get the 2 integrals ∫(1/2)dx and -(3/2)∫1/(2x+1)dx
Where do (1/2)dx and -(3/2) come from?
And where does (3/4) come...
Homework Statement
Need to refresh my memory :-S
Indefinite integral of x/(1+x^2) ..
Homework Equations
The Attempt at a Solution
Would I use partial fractions on that bad boy?
Homework Statement
\int \sqrt{tanx} dx
The Attempt at a Solution
I used the substitution u = sqrt(tanx), then x = arctan(x^2)
so:
2 \int \frac{u^{2}}{u^{4} + 1} du = 2 \int \frac{u^{2}}{(u^{2} + 1)^{2} - 2u^{2}} =2 \int \frac{u^{2}}{(u^2 -\sqrt{2}u + 1)(u^2 +\sqrt{2}u + 1)}
now...
Partial fractions of (-2x2+10x+8)/[x2(x+2)]
I initally thought that it was A/x + Bx+C/x2 + D/x+2 but you really just do Ax+B/x2 + C/x+2 ...can anyone explain why the "x2" isn't split?
I'm trying to do a question that requires the expansion of the following using partial fractions:
f(z)=\frac{1}{(1+z^3)^2}.
The fact that the bottom is squared is throwing me off for some reason... I've factorized the bottom, but I'm not sure whether I should use the complex roots or not, or...
Hi, had to learn partial fractions last year for laplace transforms, but have forgotten the general rules, and now i can't work out how to turn this into partial fractions:
\frac{s}{\left(s^{2}+4\right)\left(s^{2}+9\right)}...
Homework Statement
This problem is killing me.
I need to bust this thing up using partial fractions.
Homework EquationsThe Attempt at a Solution
I'm leaning towards it being separated like this. Is this correct?
If it is, I'm not exactly sure what I'm supposed to do next.
Homework Statement
Sorry I don't have equation editor working
1/(z+1)(z2 + 2z + 2)
Homework Equations
The Attempt at a Solution
(z2 + 2z + 2)
z2 + 1) can be factor as (z - i)(z + i) However, I'm having trouble seeing the pattern on what (z2 + 2z + 2) would become, I...
how do you integrate 1/(x2 + 1)2 ?
i have tried integration by partial fractions but when you set 1 equal to (Ax +B)(x2+1) + (Cx + D) this leads to A=B=C=0 and D=1 which just gives you the original equation
Hi
Can anybody help me with these 3 problems?:
1)
Express (3x-1)/(x+3)^2 in the form A/(x+3) + B/((X+3)^2) where A and B are constants.
2)
A curve C has parametric equations:
x=cost and y=2-cos2t (between 0 and pi)
a)prove this can be expressed as the cartesian equation y=3-2x^2
b)...
I'm sure this is a no brainer to someone, but here it is..
what is does the partial fraction of this look like in expanded form? Or how can I make it fit on the table of laplace transforms?
__(2s+1)__
(s-1)^2 + 1
Homework Statement
I need to integrate this using partial fractions. "b/(x^2-a^2)"
Homework Equations
The Attempt at a Solution
I have no idea where to begin.
Integration with partial fractions -- help!
Homework Statement
Here is the problem: http://img130.imageshack.us/img130/1673/integralthing.png
The answer should be 4.
Homework Equations
N/A
The Attempt at a Solution
Here are my steps so far:
(5x^2 - 17x + 10) / ((x-1)^3 *...
Homework Statement
\int \frac{xdx}{x^3-1}
Homework Equations
The Attempt at a Solution
Having difficulties with this one.
I managed to break it down to two partial fractions, being x-1 and x^2+x+1 but couldn't make anything out of it.
Homework Statement
Integral(sinx(x)dx/(cos^2(x)+cos(x)-2)
Homework Equations
The Attempt at a Solution
What I tried to do first was factor the denominator, so i got (cos(x)-1)(cos(x)+2)
from there, I set up my partial fractions equation trying to solve B(cos(x)-1) + A(cos(x)+2) =...
Homework Statement
expand by partial fractions:
Homework Equations
2(s+5)/(1.25*s^2+3s+9)
The Attempt at a Solution
ok I initially used the quadratic formula to get the two roots for the denominator
these being
(s+6/5+12i/5)(s+6/5-12i/5) i.e complex numbers
so now the partial fractions...
Homework Statement
\frac{dP}{dt}=P-P^{2}
It seems that Partial Fractions should be used to solve this D.E., but I cannot find an example to go by.
I even tried to rewrite the equation as:
\frac{d}{dx}Y\left(x\right)=Y\left(x\right)-Y\left(x\right)^{2}
But, that isn't helping me...
Homework Statement
Compute the integral:
int ((1-x^2)/(x^3+x)) dx
Homework Equations
int ((1-x^2)/(x^3+x)) dx
The Attempt at a Solution
I think I should use the partial fraction method to simplify the fraction
so
(1-x^2)/(x^3+x)= A/x + B/ (1+x^2)
Therefore...
Here's (what I think is) a step in partial fractions that I don't understand:
http://apthtml.com/images/partialfrac.png
I'm taking the regular partial fractions steps and I keep ending up with 1 / (1 - x) rather than 1 / (x - 1). What am I doing wrong?
Hello :smile:
I've been stuck on this question for almost 3 hours now, and I still have no idea what to do. We haven't done a question like this in class, although we have done integration with partial fractions.
Homework Statement
Evaluate...
Homework Statement
I need to integrate (2x-5)/(x^2+5x+11)
Homework Equations
The Attempt at a Solution
My problem is just finding a formula for an irreducable quadratic. I know if the denominator was x(x^2+1), I would use A/x+(Bx+C)/(x^2+1). I just don't know the formula in this...
The problem is:
((x^3)+x)/(x-1)
And i need to break it into partial fractions...
I tried long division and got:
((x^2) +x )
But the book gives me the answer of:
(x^2)+x+2+(2/(x-1))
Any help would be very much appreciated, thanks.
Homework Statement
Verify that, for all positive values of n,
1/(n+2)(2n+3) -1/((n+3)(2n+5))=(4n+9)/((n+2)(n+3)(2n+3)(2n+5))
For the series
∑_(n=0)^N▒(4n+9)/((n+2)(n+3)(2n+3)(2n+5))
Find
The sum to N terms,
The sum to infinity.
Homework Equations
no
The Attempt at a...
Homework Statement
I((3x^2+x+4)/(x^4+3x^2+2),x)
I((3x^2+x+4)/((x^2+1)(x^2+2)),x)
I(3x^2/((x^2+1)(x^2+2)),x)+I((x+4)/((x^2+1)(x^2+2)),x)
from here i have used partial fractions with no luck
Homework Equations
The Attempt at a Solution
Homework Statement
partial fractions
(-x^2+x+3)/((x-2)^2(x+1)) = A/(x-2) + B/((x-2)^2) + C/(x+1)
is my partial fractions set up right
A= -1/9 B = 1/3 C=1/9
I'm having trouble understanding what the numerator needs to be in the partial fractions.
e.g.
\frac{1}{(x-1)(x-2)^2}\equiv \frac{A}{x-1}+\frac{Bx+C}{(x-2)^2}
Notice how the first numerator has a constant A, while the second is linear Bx+C.
Actually... just now I think I may understand...
Hey guys, I have a quick question concerning partial fractions, say i have the values A(..)+B(..)+C(..) and i can only get A b substituting a value for x, but I can not make A or C go zero or A and B go to zero at the same time, if you understand ? How do I get the values B and C ? I remember...
Is it possible to use partial fractions on exponential factors? An expression like this for example...
3^x
______________________
(2^x + 3^x)(5^x + 6^x)
Would it break down into something like this?
A^x/(2^x + 3^x) + B^x/(5^x + 6^x)
What are the rules for repeated factors...
Homework Statement
I need to intergrate the following
(4x2+2x-1)/ (x3+x2)
How to set up problem using linear factors?
The Attempt at a Solution
Factoring the denominator I get:
x2(x+1)
By linear/quadratic factoring I get:
A/x + Bx+C/ x2 + d/(x+1)
Is this right?
Homework Statement
use the method of partial fractions on \int \frac{36}{(x-2)(x-1)^2(x+1)^2} dx
[b]2. Homework Equations
[b]3. The Attempt at a Solution
\frac{A}{(x-2)} + \frac{B}{(x-1)} + \frac{C}{(x-1)^2} + \frac{D}{(x+1)} + \frac{E}{(x+1)^2}
I took the 36 out of the...
Homework Statement
∫1/[(x+a)(x+b)]dx
answer is 1/(a-b) ln[(x+b)/(x+a)] + C
Homework Equations
The Attempt at a Solution
1=A/(x+a) + B/(x+b)
1=B(x+a) + A(x+b)
1=Bx+ Ba + Ax +Ab
so 0=Bx+ Ax, 1=Ba+Ab
A=-B, 1=B(a-b)
∫-1/(x+a) +∫1/(x+b)
Im not sure if I am headed in...
Homework Statement
Integrate: (x-1) / (X^2 - 4x +5)
The attempt at a solution
Normally I would try to factor this into something like (x-1) (x+3) (That's an example completely unrelated to this problem.)
However, as no easy factors quickly occurred to me I did a run through of the...