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
(
x
)
g
(
x
)
,
{\displaystyle \textstyle {\frac {f(x)}{g(x)}},}
where f and g are polynomials, is its expression as
f
(
x
)
g
(
x
)
=
p
(
x
)
+
∑
j
f
j
(
x
)
g
j
(
x
)
{\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
F(X)=[tex]\int[/\frac{1}{1+t^3}
Homework Equations
The Attempt at a Solution
I have tried different substitutions to find fog where g(t) = ? But am getting stuck
Homework Statement
Actually i want to ask something actually very easy...
i just don't know the meaning of some words in different questions...
firstly... multiplication of A and B means A*B right?
how about multiplication of A by B means A*B or A/B??
secondly... division of A by B means...
Homework Statement
Evaluate the integral: integral (-17e^x-36)/(e^(2x)+5e^x+6 dx
Homework Equations
partial fractions
The Attempt at a Solution
Basically, what i did was factored the bottom into (e^x+2) and (e^x+3) because when i expand that, it equals the bottom. From there, i...
Homework Statement
I am supposed to evaluate the integral using partial fractions.
\int \frac{1}{(x+5)^2(x-1)} dx
2. The attempt at a solution
So after doing all the work, I get
(-1/36)ln|x+5| - (13/6)ln|x+5| + (1/36)ln|x-1|
But the answer in the book appears as
(-1/36)ln|x+5| -...
Homework Statement
1/((x^2-1)^2)
Homework Equations
The Attempt at a Solution
so i get (Ax+B)/(x^2-1) + (Cx+D)/((x^2-1)^2)
then i multiply both sides by ((x^2-1)^2)
then i get 1=(Ax+B)(x^2-1)+ (Cx+D)
then i multiply it out Ax^3+Bx^2 -Ax +Cx +D =1
then i equate...
Homework Statement
(3x^2-4)/(x^3-4x-6)
Homework Equations
I guess integration by parts... But how do i set this up?
The Attempt at a Solution
The numerator is exponentially lower than the denominator, so no long division.
The denominator seems not to factor out into anything...
Homework Statement
\int\frac{e^x}{(e^x-2)(e^2x +1)} it should be e to the power of 2x
Homework Equations
Using substitution u=e^x, and then using partial fractions
The Attempt at a Solution
I have done this problem two separate ways. One with substitution and then partial...
Homework Statement
Solve the integral x/x^2+4x+13
Homework Equations
I think that you would use partial fractions but I'm not really sure. I know that you need to complete the square on the denominator.
The Attempt at a Solution
The completed square would be (x+2)^2+9. I don't...
Homework Statement
Integrate
x^3 + 49 / x^2 + 5x + 4
Homework Equations
The Attempt at a Solution
Since the numerator has an x cubed, but the denominator only has an x squared, I know I need to divide the numerator by something.
I'm not sure what, but maybe the...
Homework Statement
The problem asks to evaluate the integral using partial fractions, but I just cannot find out which trick to get this one to work. the equation is
\int\frac{x^3+x^2+2x+1}{(x^2+1)(x^2+2)}
The Attempt at a Solution
I have tried setting it up as a partial fraction...
Homework Statement
integrate (4x^2 + 3x + 6)/x^2 (x+2) dx
Homework Equations
don't have sorry..
The Attempt at a Solution
firstly = A/x + B/x^2 + C /x+2 , = A(x^2)(x+2) + B(x)(x+2) + C(x)(x^2) equating with the 4x^2 + 3x + 6,then i integrate it,but my ans turn out to be...
I've been working with Laplace Transforms and integration ALOT lately. Many times I windup having to use partial fractions to solve the problem and frankly my algebra skills just aren't up to the task.
Take this fraction for example;
I know 3 ways to do it... 1 of the ways doesn't work unless...
\frac{s-1}{s(s-2)^2}
How can I expand this fraction?
\frac{A}{s} + \frac{B}{(s-2)} + \frac{C}{(s-2)^2}
right?
This gives me the equation
As^3 - 6As^2 + 12As - 8A Bs^3 - 4Bs^2 + 4Bs + Cs^2 - 2Cs = s-1
so that
(1) A + B =0
(2)- 6A - 4B + C = 0
(3) 12A + 4B - 2C = 1
(4)...
\int e^{ax}cosbx
This one is driving me insane.
So I used e^ax as u and cosbx dx as dv. And then I did it again using e^ax as u and sinbx as dv which left me with \int e^{ax}cosbx = \frac{1}{b}e^{ax}sinbx + \frac{a}{b^{2}}e^{ax}cosbx - \frac{a^{2}}{b^{2}}\int e^{ax}cosbxdx
I have no...
Homework Statement
How does one integrate e.g. \frac{1+x}{(2+x)^{3/2}} by partial fractions?
The Attempt at a Solution
I have no idea about this. I've never seen this technique applied with fractional powers before.
Homework Statement
Derive a formula for the antiderivative of sec x using the identity that sec x= cos x/ (1-sin^2x). Use a substitution for sin x and then partial fractions. Then multiply the solution by (1+sin x)/ (1+sin x) to obtain the more familiar formula for the antiderivative...
This is probably a "basic" question, but I can't seem to remember how to do partial fractions problems where there is only a 1 in the numerator.
For example (just making this up), let's say I have:
1/s(s+4)(s+5)
So what I'd do is 1/s(s+4)(s+5) = A/s + B/(s+4) + C/(s+5) as one would expect...
I'm trying to solve this integral but I'm not sure if I'm on the right track. My question is: can this integral be solved by partial fractions decomposition? I solved the problem that way but I'm not sure if it is the right answer. thanks!
∫1-x+2x^2-x^3 ÷ x(x^2+1)^2
Homework Statement
\int\frac{dx}{x(1+ln x)}
Homework Equations
Partial Fractions? Maybe I am solving this wrong...
The Attempt at a Solution
\frac{A}{X} + \frac{B}{1+ln x} = 1
A(1+lnx) + Bx =1
A + Alnx + Bx =1
This doesn't seem to work out properly. I have been having a...
Homework Statement
\int {\frac{{2s + 2}}
{{(s^2 + 1)(s - 1)^3 }}ds}
The Attempt at a Solution
This is a long one...First, I split the integrand into partial fractions and find the coefficients:
\begin{gathered}
\frac{{2s + 2}}
{{(s^2 + 1)(s - 1)^3 }} = \frac{{As + B}}...
Homework Statement
\int {\frac{{2s + 2}}
{{(s^2 + 1)(s - 1)^3 }}ds}
The Attempt at a Solution
This is a long one...First, I split the integrand into partial fractions and find the coefficients:
\begin{gathered}
\frac{{2s + 2}}
{{(s^2 + 1)(s - 1)^3 }} = \frac{{As + B}}...
Homework Statement
Hi everyone, here is a new partial fractions question I just cannot understand:
\int\frac{x^{3}}{x^{3}+1}dx
Homework Equations
Partial Fractions, difference of perfect cubes, polynomial long division
The Attempt at a Solution
\int\frac{x^{3}}{x^{3}+1} dx...
Homework Statement
\[
\int {\frac{{e^t dt}}
{{e^{2t} + 3e^t + 2}}}
\]
I'm not quite sure how to start this one...Any hints? I tried bringing e^t down to the denominator and multiplying it out which still didn't help. I can't see a way to factor the denominator or split this into a...
Homework Statement
∫1/ x^3-1 dx, ok how would i do this
Homework Equations
∫dx/ x^2+a^2= 1/a tan^-1 (x/a) +c
i tried to simplify x^3-1 = (x+1)(x-1)(x+1)
got an exam coming up in a few days and half way through my question i ran into a partial fractions question instead of having the standard (1/(y+c)(y+d))= A/(y+c) + B(y+d) and multiplying out i had a double root so (1/(y+c)(y+c)) does this change the way i go about the question and are there...
.. oy, I'm just not sure how to find 3 constants!
Here is my problem:
5x^2-4/(x-2)(x+2)(x-1) = A/(x-2)+B/(x+2)+C/(x-1)
.. i got a bit of it done, but it's all wrong
OH! and what am i supposed to do if the numerator of the first equation does not have any sort of variable with it??
my...
For a rational function, (x^2+1)/(x^2-1) = (x^2+1)/[(x+1)(x-1)], if we were to split it into partial fractions so that (x^2+1)/(x^2-1) = A/(x+1) + B/(x-1) = [A(x-1) + B(x+1)]/(x^2-1)...solving for A and B get us A = -1 and B = 1. This would mean that (x^2+1)/(x^2-1) = 2/(x^2-1)...which doesn't...
I'm a little mixed up on the integration for partial fraction decomposition.
I basically have x/ x(x^2 + 1)
I'm wondering for the (x^2 + 1) part, am I to put Ax + B over it because it is a raised power, or since the outside bracket is not squared, it is to only have one variable over it.
Homework Statement
we have 4/((s^2) + 4)(s-1)(s+3)
Homework Equations
The Attempt at a Solution
dividing it up do we get:
A/((s^2) + 4) + B/(s-1) + C/(s+3) = 4
or is it
(As + B)/((s^2) + 4) + C/(s-1) + D/(s+3) = 4
Homework Statement
[e^(-2s)] / (s^2+s-2)
Find the inverse Laplace transform.
Homework Equations
The Attempt at a Solution
I know that I can factor the denominator into (s+2)(s-1). Then I tried to use partial fractions to split up the denominator, but I don't know how to do that...
Telescoping Method & Partial Fractions...PLEASE HELP!
Homework Statement
Find the sum of the series from n=1 to infinity...
2/(4n^2-1)
Homework Equations
The Attempt at a Solution
I want to use the telescoping method...
2/(4n^2) = 2/[(2n-2) * (2n+1)]
I am following an...
Hello all,
I've got an exam tomorrow so any quick responses would be appreciated. I'm following the Boas section on Laurent series... Anyway, here's my problem:
In an example Boas starts with f(z) = 12/(z(2-z)(1+z), and then using partial fractions arrives at f(z) = (4/z)(1/(1+z) +...
(t+1) dx/dt = x^2 + 1 (t > -1), x(0) = pi/4
I have attempted to work this by placing like terms on either side and then integrating.
1/(x^2 + 1) dx = 1/(t + 1) dt
arctan x = ln |t + 1| + C
x = tan (ln |t + 1|) + C
pi/4 = tan(ln |0 + 1|) + C
pi/4 = C
x = tan (ln |t + 1|)...
Homework Statement
Solve y"+4y'=sin 3t subject to y(0)=y'(0)=0 using Laplace Transform
The Attempt at a Solution
So I got:
s^2Y(s)-sy(0)-y'(0)+4[sY(s)-y(0)]=\frac{3}{s^2+9}
\Rightarrow Y(s)=\frac{3}{(s^2+9)(s^2+4)}
Now it looks like two irreducible quadratics, which I...
Homework Statement
Evaluate the indefinite integral.
int (6 x + 7)/(x^2 + 1) dx `
The Attempt at a Solution
A/(x + 1) + B/(x - 1)
6x + 7 = A(x - 1) + B(x + 1)
6x + 7 = (A + B)x + (-A + B)
A + B = 6
-A + B = 7
A + (7 + A) = 6
2A = -1.
A = -.5
B = 3.5
So the...
[SOLVED] integration by partial fractions
Homework Statement
\int((2x^2-1)/(4x-1)(x^2+1))dx
Homework Equations
A1/ax+b + A2/(ax+b)^2 + ... + An/(ax+b)^n
The Attempt at a Solution
(2x^2-1)/(4x-1)(x^2+1) = A/4x-1 + Bx+C/x^2+1
2x^2-1/x^2+1 = A + Bx+C(4x-1)/x^2+1
set x =...
Integrate using partial fractions:
(int) (x^3)/(x^2 -1) dx
I have put into the form (int) (x^3)/((x-1)(x+1)) dx
I thought partial fractions had this property:
'Partial fractions can only be done if the degree of the numerator is strictly less than the degree of the denominator.'...
Homework Statement
Integration of 1/(x^2-5x+6)
Homework Equations
The Attempt at a Solution
I know i cannot do ln|x^2-5x+6|
I've tried some form of substitution or intergration by parts, and they don't work.
Should I factor the bottom?
Homework Statement
The problem is from Stewart, Appendix G, A58, no.45.
Suppose that F, G, and Q are polynomials, and:
F(x)/Q(x) = G(x)/Q(x)
for all x except when Q(x) = 0. Prove that F(x) = G(x) for all x. [Hint: Use Continuity]
The Attempt at a Solution
I thought the statement was...
Hi!
There's this one problem that I'm having troubles with. I've tried using the decomposition method, but I've ended up getting a messy answer. If someone can give me tips or the solution to the problem, I'll appreciate it. Here's the problem: solve the integral of 1/ y^2-1 dx.
Homework Statement
http://books.google.com/books?id=qFNZIUQ_MYUC&pg=PA142&lpg=PA142&dq=loren+larsen+%224.3+23%22&source=web&ots=YKlIl_yPb3&sig=MC2QCtuBii9za-vd4FkAJadZ_dI
I am working on 4.3.23b)
I can get that
\sum_{i=1}^n \frac{g(x_i)}{f'(x_i)}\frac{1}{x-x_i} = g(x)/f(x)
but I do not...
Check out the link to see the problem and my work. Can anyone see where I made an error? I can't seem to figure out where I went wrong.
https://www.physicsforums.com/attachment.php?attachmentid=11519&stc=1&d=1194912644"
Evaluate the integral of x^2-x/(x^2-1)^2 from 0 to 1.
* I know that I have to use partial fractions in order to make the integral integratable.
My attempt at partial fractions:
A/(x-1) + (B/(x+1)) + (Cx+D/(x^2-1)^2)
Is this setup right? (Once I have it set up correctly, I know how...
Homework Statement
I am given 9/[(s-1)(s-1)(s-4)] as part of a Laplace Transform. I'm supposed to decompose into partial fractions.
Homework Equations
So 9/[(s-1)(s-1)(s-4)]= D/(s-1)+E/(s-1)+F/(s-4)
The Attempt at a Solution
To simplify:
9= D(s-1)(s-4)+ E(s-1)(s-4)+ F(s-1)^2...
Arc Length, Irreducible quadratic factors
i'm having a hard time seeing this method, and i have to use this method on one of the problems I'm doing to find it's Arc Length.
L=\int_{\sqrt{2}}^{\sqrt{1+e^{2}}}\frac{v^{2}dv}{v^{2}-1}}
the book suggests to first divide then use a...
Homework Statement
I don't understand something I have read about partial fractions so I wonder if anyone can help!
To each repeated linear factor in the denominator of the form (x-a)^2, there correspond partial fractions of the form : A/(x-a) + B/(x-a)^2
Is this true if we have...