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Rational functions

Repeated linear factors

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

A1x−a+A2(x−a)2+⋯+Am(x−a)m\frac{A_1}{x-a}+\frac{A_2}{(x-a)^2}+\cdots+\frac{A_m}{(x-a)^m}

A factor raised to m requires every power from 1 through m. Solve the constants together. Power 1 yields a logarithm; the remaining powers use the power rule with negative exponents.

Common mistake

Do not omit intermediate-power terms.

DEVELOPED GUIDE · 9 min

Decompose a repeated linear factor without omitting any power.

When to choose this method

  • The integrand is a polynomial quotient with a denominator containing a power of (x−a).
  • Compare degrees first. Divide before decomposing if the numerator degree is at least as high.

Before calculating

  1. Include a fraction for every power, from 1 to the factor multiplicity.
  2. Clear denominators, compare coefficients and integrate the resulting terms.

A complete example, step by step

∫2x+3(x−1)2 dx\int\frac{2x+3}{(x-1)^2}\,dx

  1. The denominator has degree 2 and the numerator degree 1. The fraction is proper; x=1 is excluded.

    deg⁡P=1<2=deg⁡Q,x≠1\deg P=1<2=\deg Q,\qquad x\ne1

  2. The factor occurs twice. We need a term with (x−1) and another with (x−1)².

    2x+3(x−1)2=Ax−1+B(x−1)2\frac{2x+3}{(x-1)^2}=\frac A{x-1}+\frac B{(x-1)^2}

  3. Multiply by the common denominator; we now compare polynomials.

    2x+3=A(x−1)+B=Ax+(B−A)2x+3=A(x-1)+B=Ax+(B-A)

  4. The x coefficients give A=2. Constant terms give B−A=3, hence B=5.

    A=2,B−A=3⟹B=5A=2,\quad B-A=3\quad\Longrightarrow\quad B=5

  5. Split the integral. The first term is logarithmic; the second is a power with exponent −2.

    I=2∫dxx−1+5∫(x−1)−2 dxI=2\int\frac{dx}{x-1}+5\int(x-1)^{-2}\,dx

  6. Integrate, retain the absolute value and add one constant.

    I=2ln⁡∣x−1∣−5x−1+CI=2\ln|x-1|-\frac5{x-1}+C

Check the result and domain

The derivative recovers the original quotient. Interpret the primitive separately on x<1 and x>1.

2x−1+5(x−1)2=2x+3(x−1)2\frac2{x-1}+\frac5{(x-1)^2}=\frac{2x+3}{(x-1)^2}
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Now try it yourself

Solve on paper before opening the hint or answer. These are self-assessment activities; to check a typed answer, open Learn.

EXERCISE 1∫dx(x−2)2\int\frac{dx}{(x-2)^2}
Show a hint

This is power −2 after shifting x.

Check my result−1x−2+C-\frac1{x-2}+C
EXERCISE 2∫3x+1(x+1)2 dx\int\frac{3x+1}{(x+1)^2}\,dx
Show a hint

Write 3x+1=3(x+1)−2.

Check my result3ln⁡∣x+1∣+2x+1+C3\ln|x+1|+\frac2{x+1}+C
EXERCISE 3∫x+2(x−1)3 dx\int\frac{x+2}{(x-1)^3}\,dx
Show a hint

Write x+2=(x−1)+3.

Check my result−1x−1−32(x−1)2+C-\frac1{x-1}-\frac3{2(x-1)^2}+C
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Connect with other methods

Partial fractions →Equal or higher degree: divide first →Repeated quadratics and mixed factors →
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