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




{\displaystyle \textstyle {\frac {f(x)}{g(x)}},}

where f and g are polynomials, is its expression as











{\displaystyle {\frac {f(x)}{g(x)}}=p(x)+\sum _{j}{\frac {f_{j}(x)}{g_{j}(x)}}}
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.

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  1. Z

    Partial fractions with complex linear terms

    I am interested specifically in solving this problem by factoring the quadratic term into complex linear factors. $$s^2+4=0$$ $$\implies s=\pm 2i$$ $$\frac{5s+6}{(s-2i)(s+2i)(s-2)}=\frac{A}{s-2i}+\frac{B}{s+2i}+\frac{C}{s-2}$$ We can solve for ##C## using the cover-up method with ##s=2## to...
  2. C

    I Using Residues (Complex Analysis) to compute partial fractions

    Dear Everybody, I am wondering how to compute the partial fraction decomposition of the following rational function: ##f(z)=\frac{z+2}{(z+1)^2(z^2+1)}.## I understand how to do the simple poles of the function and how it is related to the decomposition's constants, i.e...
  3. chwala

    Solve the given problem that involves integration

    For part (a), Using partial fractions (repeated factor), i have... ##7e^x -8 = A(e^x-2)+B## ##A=7## ##-2A+B=-8, ⇒B=6## $$\int {\frac{7e^x-8}{(e^x-2)^2}}dx=\int \left[{\frac{7}{e^x-2}}+{\frac{6}{(e^x-2)^2}}\right]dx$$ ##u=e^x-2## ##du=e^x dx## ##dx=\dfrac{du}{e^x}## ... also ##u=e^x-2##...
  4. chwala

    Solve the problem that involves partial fractions

    Let $$y=\frac {1+3x^2}{(1+x)^2(1-x)}= \frac {A}{1-x}+\frac {B}{1+x}+\frac {C}{(1+x)^2}$$ $$⇒1+3x^2=A(1+x)^2+B(1-x^2)+C(1-x)$$ $$⇒A-B=3$$ $$2A-C=0$$ $$A+B+C=1$$ On solving the simultaneous equations, we get ##A=1##, ##B=-2## and ##C=2## therefore we shall have, $$y=\frac {1}{1-x}+\frac...
  5. anita chandra

    A Does this integration have a closed form solution?

    I was trying to solve a differential equation that I defined to study the dynamics of a system. Meanwhile, I encounter integration. The integration is shown in the image below. I tried some solutions but I am failed to get a solution. In one solution, I took "x" common from the denominator terms...
  6. E

    Integrating by Partial Fractions

    I was doing this problem from Griffith's electrodynamics book and can't figure out how to do this integral. The author suggested partial fractions but the denominator has a fractional exponent which I have never seen for partial fractions, and so, I am unsure how to proceed. The integral I am...
  7. A

    I Equating coefficients of complex exponentials

    I have an equation that looks like ##i\dot{\psi_n}=X~\psi_n+\frac{C~\psi_n+D~a~\psi^\ast_{n+1}+E~b~\psi_{n+1}}{1+\beta~(D~\psi^\ast_{n+1}+E~\psi_{n+1})}## where ##E,b,D,a,C,X## are constants. I have the ansatz ##\psi_n=A_n~e^{ixt}+B^\ast_n~e^{-itx^\ast}##, ##x## and ##A_n,B_n## are complex...
  8. Jeviah

    How to get the third value (A), using partial fractions

    Homework Statement y(w)= 3/(iw-1)^2(-4+iw) Homework Equations N/A The Attempt at a Solution 3/(iw-1)^2(-4+iw) = A/iw-1 + B/(iw-1)^2 + C/-4+iw for B iw = 1 B=3/-4+1 = -1 for C iw = 4 C= 3/(4-1)^2 = 1/3 I know the answer for A should be -1/3 however I am unsure how to obtain this as if the...
  9. F

    I Partial Fractions: Explained

    im a bit confused about partial fractions If we have something like x/((x+1)(x+2)) we could decompose it into a/(x+1) +b/(x+2) If we had something like x/(x+1)^2 we could decompose it into a/(x+1) + b/(x+1)^2 We use a different procedure when there is a square in part of the polynomial in...
  10. karush

    MHB 242 .10.09.8 Express the integrand as a sum of partial fractions and evaluate integral

    $\tiny{242 .10.09.8}\\$ $\textsf{Express the integrand as a sum of partial fractions and evaluate integral}$ \begin{align*}\displaystyle I&=\int f \, dx = \int\frac{\sqrt{16+5x}}{x} \, dx \end{align*} \begin{align*}\displaystyle f&=\frac{\sqrt{16+5x}}{x}...
  11. A

    MHB Partial fractions ( part of a logistic equation)

    Hi everyone, I am stuck on a problem. I need to give a partial fraction of 1/N(k-N). I have tried every method so far ( plotting roots, systems of equations). I think I found A=1/k but I have no clue how to find B value. I would really appreciate any help as I am a desperate student trying to...
  12. karush

    MHB partial fractions?

    $\tiny{}$ \begin{align*} \displaystyle I_{88}&=\int\frac{1}{(x+2)\sqrt{x^2+4x+3}} \, dx \\ &=? \end{align*} would partial fractions be best for this?
  13. M

    MHB Partial fractions (5x^2+1)/[(3x+2)(x^2+3)]

    Trouble here in the below partial fraction (Bug) $\frac{5x^2+1}{(3x+2)(x^2+3)}$ One factor in the denominator is a quadratic expression Split this into two parts A&B $\frac{5x^2+1}{(3x+2)(x^2+3)}=\frac{A}{(3x+2)}+\frac{Bx+c}{(x^2+3)}$...
  14. karush

    MHB -7.4.89 partial fractions

    partial fractions $$\int\frac{3x^2+x+12}{(x^2+5)(x-3)} =\frac{A}{(x^2+5)}+\frac{B}{(x-3)}$$ $$3x^2+x+12=A(x-3)+B(x^2+5)$$ x=3 then 27+3+12=14B 3=B x=0 then 12=-3A+15 1=A $$\int\frac{1}{(x^2+5)} \, dx +3\int\frac{1}{(x-3)}\, dx$$ $\displaystyle...
  15. karush

    MHB 242t.8.5.9 expand the quotient by partial fractions

    $\tiny{242t.8.5.9}$ $\textsf{expand the quotient by}$ $\textbf{ partial fractions}$ \begin{align*}\displaystyle y&=\int\frac{dx}{9-25x^2} &\tiny{(1)}\\ \end{align*} $\textit{expand and multiply every term by $(3+5x)(3-5x)$}$ \begin{align*}\displaystyle...
  16. dykuma

    Convert Partial Fractions & Taylor Series: Solving Complex Equations

    Homework Statement and the solution (just to check my work) Homework Equations None specifically. There seems to be many ways to solve these problems, but the one used in class seemed to be partial fractions and Taylor series. The Attempt at a Solution The first step seems to be expanding...
  17. karush

    MHB Express the integrand as sum of partial fractions

    $\tiny{}$ $\textsf{Express the integrand as sum of partial fractions}$ \begin{align} && I_{49}&=\int\frac{30s+30}{(s^2+1)(s-1)^3}\, ds& &(1)& \\ &\textsf{expand}& \\ && &=\displaystyle 15\int\frac{1}{(s^2+1)}\, ds -15\int\frac{1}{(s-1)^2}\, ds +30\int\frac{1}{(s-1)^3}\, ds&...
  18. S

    MHB Partial Fractions: Struggling to Remember? Help Here!

    struggling to remember anything about partial fractions, can anybody help me with this? 6x-5 (x-4) (x²+3)
  19. karush

    MHB 206.5.64 integral by partial fractions

    $\textbf{206.5.64 integral by partial fractions} \\ \displaystyle I_{64}= \int\frac{9x^3-6x+4}{x^3-x^2} \, dx \\ \text{expand} \\ \displaystyle \frac{9x^3-6x+4}{x^3-x^2} = \frac{9(x^3-x^2)+9x^2+6x+4}{x^3-x^2} = 9 + \frac{9x^2+6x+4}{x^2(x-1)} \\ \textbf{stuck!}$
  20. P

    MHB Sava's question via email about integration with partial fractions.

    As there is a repeated root, the partial fraction decomposition we should use is: $\displaystyle \begin{align*} \frac{A}{x - 1} + \frac{B}{\left( x - 1 \right) ^2 } + \frac{C}{x - 2} &\equiv \frac{x^2}{\left( x - 1 \right) ^2\,\left( x - 2 \right) } \\ \frac{A\,\left( x - 1 \right) \left( x - 2...
  21. Brandon Trabucco

    B Complex Integration By Partial Fractions

    Hello, I am enrolled in calculus 2. Just having started a section in our textbook about integration by partial fractions, I eagerly began trying to use this integration technique wherever I could. After messing around for multiple days, I ran into this problem: ∫ 1/(x^2+1)dx I immediately...
  22. AntSC

    Partial Fractions with Ugly Coefficients

    Homework Statement The question is stated at the top of the attached picture with a solution 20160303_095831.jpg The correct results of the coefficients are A=2, B=-5, C=1 I have tried this problem multiple times and am still getting ugly coefficients. I have no idea why. A fresh pair of eyes...
  23. kostoglotov

    Why is my partial fraction decomp. wrong?

    Homework Statement Decompose \frac{2(1-2x^2)}{x(1-x^2)} I get A = 2, B =-1, C = 1, but this doesn't recompose into the correct equation, and the calculators for partial fraction decomposition online all agree that it should be A = 2, B = 1, C = 1. Here is one of the online calculator results...
  24. Mark44

    Insights Partial Fractions Decomposition - Comments

    Mark44 submitted a new PF Insights post Partial Fractions Decomposition Continue reading the Original PF Insights Post.
  25. T

    Math problem integration by partial fractions

    Homework Statement integrate (4x+3)/(x^2+4x+5)^2 Homework EquationsThe Attempt at a Solution I know to solve this problem you have to work with partial fractions, in the solution we were given they solve as followed 4x+3=A(x^2+4x+5)'+B I don't know why they take the derivative of x^2+4x+5...
  26. King_Silver

    Method of Partial Fractions integral help

    I have a question where f(x) = 20-2x^2/(x-1)(x+2)^2 and have solved for constants A,B and C. A = 2 B = -4 C = -4 I have worked this out myself. Now I am told to compute the indefinite integral and I am getting this answer but apparently it is wrong and I don't understand how? My answer...
  27. O

    Definite integral involving partial fractions

    Homework Statement Homework Equations trigonometric identities The Attempt at a Solution I did a trig substitution of u=tan(θ/2) and from that I could substitute cos(θ) = 1-u2/1+u2 dθ = 2/(1+u2) du = 1/2 sec2(θ/2) dθ I simplified a bit and changed the bounds to get 2du/(5u2 + 1)(1 + u2)2...
  28. ognik

    MHB Cauchy Integral Theorem with partial fractions

    (Wish there was a solutions manual...). Please check my workings below Show $ \int \frac{dz}{{z}^{2} + z} = 0 $ by separating integrand into partial fractions and applying Cauchy's Integral theorem for multiply connected regions. For 2 paths (i) |z| = R > 1 (ii) A square with corners $ \pm 2...
  29. DameLight

    "Partial Fractions" Decomposition Integrals

    Hello, I was just introduced to this concept and I have solved a few problems, but I haven't come across any with denominators to a raised power yet. ∫ 1 / [(x+7)(x^2+4)] dx I would appreciate any directed help. 1. from the initial state I have broken the fraction into two assuming that...
  30. J

    Very long Taylor expansion/partial fraction decomposition

    Homework Statement I want to express the following expression in its Taylor expansion about x = 0: $$ F(x) = \frac{x^{15}}{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)} $$ The Attempt at a Solution First I tried to rewrite the function in partial fractions (its been quite a while since I've last...
  31. karush

    MHB Solve Int Partial Fractions: $\int\frac{6{x}^{2}+22x-23} {(2x-1)(x+3)(x-2)} dx$

    $\int\frac{6{x}^{2}+22x-23} {(2x-1)(x+3)(x-2)} dx $ Solve using partial fractions $\frac{A}{2x-1}+\frac{B}{x+3}-\frac{C}{x-2}$ I pursued got A=2 B=-1 C=-3 Then?
  32. P

    Partial Fractions: Decomposing a Rational Function

    Suppose we have a rational function ##P## defined by: $$P(x) = \frac{f(x)}{(x-a)(x-b)}$$ This is defined for all ##x##, except ##x = a## and ##x = b##. To decompose this function into partial fractions we do the following: $$\frac{f(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$$ Multiplying...
  33. StrangeCharm

    Integration by Partial Fractions Help

    Homework Statement ∫ [x^(3)+4] / [x^(2)+4] dx Homework Equations N/A The Attempt at a Solution I know that the fraction is improper, so I used long division to rewrite it as x+(-4x+4)/[x^(2)+4]. Given the form S(x)+R(x)/Q(x), Q(x) is a distinct irreducible quadratic factor [x^(2)+4]. I used...
  34. QuantumCurt

    Integral with partial fractions

    Homework Statement I'm currently in Calculus 3, and the professor gave us a "retro assignment" which is basically a bunch of tough integrals from Calculus 2. I think my process here is valid, but when I check my answer on Wolfram, they're getting a slightly different final answer...
  35. 462chevelle

    Partial fractions integral

    Homework Statement integral(0>1) of (x^2+x)/(x^2+x+1)dx Homework Equations Factor denominator, and set numerator with A,B,C, etc. multiply both sides by the common denominator. The Attempt at a Solution Since the denominator won't factor at all I don't really know where to start, I could...
  36. D

    Partial Fractions: Why Does (x+1)2(2x+1) Need 3 Terms?

    Why, when a fraction has repeated linear terms in its denominator e.g. (11x2+14x+5)/[(x+1)2(2x+1)] does it have to be split into three partial fractions, A/(x+1) + B/(x+1)2 + C/(2x+1)? When my first saw this example, my initial reaction was to split it into A/(x+1)2 +B/(2x+1), but after working...
  37. S

    Inverse laplace transform without partial fractions

    Homework Statement take inverse laplace of: 6/[s^4(s-2)^2] Homework Equations 6/[s^4(s-2)^2] The Attempt at a Solution I used partial fractions. I was wondering if It could be manipulated to where I could use the laplace table?
  38. RJLiberator

    Partial Fractions - irreducibility question

    1. x^2-x+1 Is this factorable? My initial thinking is NO. However, I can complete the square and it becomes (x-1/2)^2-3/4, but this doesn't seem to help me. Would this be considered factorable? 2. Turn 1/x^2-x+1 into partial fractions Clearly, after I answer #1 correctly, #2 will be more...
  39. Y

    How to find the partial fractions for this expression?

    Homework Statement Find the partial fractions for this expression. (((n+1)*(sqrt(n)) - n*(sqrt(n+1))) / (n*(n+1))) The Attempt at a Solution The final answer is 1/sqrt(n) - 1/(sqrt(n+1)) My work: A/n - B/(n+1) = n*sqrt(n+1) - (n+1)*(sqrt(n)) I am subbing in n = -1 and n = 0 to solve for...
  40. C

    Partial Fractions in Differential Equations

    I'm a little rusty with partial fractions, and I can't seem to find my error once I get up to that point. Homework Statement dy/dx = (y^2 - 1) / x Homework Equations The Attempt at a Solution Cross-mutliply x dy = (y^2 - 1) dx Divide by the appropriate terms dy / (y^2...
  41. A

    MHB Tricky Partial Fractions Question

    Hey guys, Here is another pair of questions that I'm doubting at the moment: I used partial fractions for A and got (Bx+C)/x^2 + Ax/(x-1)^2 + Dx(x-1) which led me to compute A=1, B=0, C= -1, and D=0, which already sounds off. Do you guys have any suggestions? Also, for 5b, I calculated B=...
  42. E

    Partial Fractions - Integration

    Homework Statement Evaluate the integral. (Remember to use ln |u| where appropriate. Use C for the constant of integration.) \int \frac {5x^2 - 20x +45}{(2x+1)(x-2)^2}\, dx Homework Equations 5x^2 - 20x +45 = 5 (x^2 -4x +9) The Attempt at a Solution I'm able to come up with an...
  43. S

    MHB Understanding Partial Fraction Decomposition in Integrals

    First the example problem. This is an integral of the whole thing (3x^3+24x^2+56x-5) / (x^2+8x+17)^2 The answer comes out to be 3/2 ln(x^2+8x+17) - (49/2 tan^-1(x+4)) - (25x+105 / 2(x^2+8x+17) + C I would show all the steps but I'm still not sure on how to use the format tools, so that...
  44. A

    MHB Quick Integral (U-substitution and partial fractions) Questions

    Hey guys, I'd really appreciate it if I could get some quick help for this problem set I'm working on. For question one, I just did a quick u substitution for x^4 and managed to get x^4 * sin(x^4)+cos(x^4) + C. For part b, I used integration by parts and took ln(4t) as u and the rest as...
  45. L

    Calculating Partial Fractions find A, B and C

    Homework Statement For the equation shown below: x2+2x+3 / (x2+9)(X-3) = Ax+B/(x2+9) + C/(x-3) Find A, B and C Homework Equations The Attempt at a Solution C = 1 B = 2 A = ? Find C which = 1 by putting x=3 and working out x2+2x+3/(x2+9), then multiply out equation...
  46. B

    Partial Fractions - 3 Unknowns

    Hello, i've come across a partial fractions problem that I don't know how to solve - Usually, the denominator of the fraction I need to split up into two separate fractions is a quadratic, but in this instance it's a cubic. Specifically, the problem I'm having is that two of the factors to...
  47. F

    Integration by Partial Fractions

    Homework Statement Find the indefinite integral of the below, using partial fractions. \frac{4x^2+6x-1}{(x+3)(2x^2-1)} Homework Equations ?The Attempt at a Solution First I want to say there is probably a much easier and quicker way to get around certain things I have done but I have just...
  48. J

    Using trig substitution or partial fractions?

    When would you use trig substitution vs. partial fractions? I know partial fractions is when you have a polynomial over a polynomial, but some of the problems in the trig substitution section in my book had polynomial over polynomial and used trig substitution?
  49. jegues

    How to Solve Partial Fractions Expansion?

    Homework Statement Find the partial fractions expansion in the following form, G(s) = \frac{1}{(s+1)(s^{2}+4)} = \frac{A}{s+1} + \frac{B}{s+j2} + \frac{B^{*}}{s-j2} Homework Equations The Attempt at a Solution I expanded things out and found the following, 1 = A(s^{2} + 4)...
  50. J

    Multiplying Partial Fractions: Understanding the Rules

    Homework Statement Homework Equations After looking through this on Wiki, I'm a little confused as to how these partial fractions are multiplied out. Is there a rule or something for this? With simpler partials I can do it but this one is something else! The Attempt at a Solution