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Section 2.4 Calculus 2

Integrating with Partial Fractions
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2.4 Integrating with Partial Fractions

2.4.1 Rational Functions and Partial Fractions

  • A function of the form \(\frac{f(x)}{g(x)}\) where \(f,g\) are both polynomials is called rational.
  • The rational function \(\frac{f(x)}{(x+r)^m}\) may be split into the partial fractions \(\frac{A_1}{x+r}+\frac{A_2}{(x+r)^2}+\dots+\frac{A_m}{(x+r)^m}\), provided the degree of the numerator is less than the denominator.
  • Example Expand \(\frac{2x^2-7x+6}{(x-2)^3}\) using partial fractions.
  • The rational function \(\frac{f(x)}{(x^2+px+q)^n}\) (where \(x^2+px+q\) is irreducible) may be split into the partial fractions \( \frac{B_1x+C_1}{x^2+px+q}+ \frac{B_2x+C_2}{(x^2+px+q)^2}+ \dots+ \frac{B_mx+C_m}{(x^2+px+q)^n} \), provided the degree of the numerator is less than the denominator.
  • Example Expand \(\frac{3x^2+2x+4}{x^4+2x^2+1}\) using partial fractions.
  • When the denominator is a product of \((x+r)^m\) and \((x^2+px+q)^n\) terms, simply sum up the appropriate partial fractions for each factor.
  • Example Describe the partial fractions which expand the rational function \(\frac{f(x)}{(x+3)^3(x^2-2x+3)^2}\).

2.4.2 Integrating Partial Fractions

  • Expanding rational functions using partial fractions allows us to integrate.
  • Example Find \(\int\frac{2x^2+5x-9}{(x-1)(x+1)(x-2)}\,dx\).
  • Example Find \(\int\frac{4y^2+14y+15}{y^3+4y^2+5y}\,dy\).
  • If the numerator has degree greater than or equal to the denominator, you will need to use long polynomial division to break down the rational function first.
  • Example Find \(\int\frac{2t^3+t^2+3t+2}{(1+t)(1+t^2)}\,dt\).

Review Exercises

  1. Expand \(\frac{4x^2+16x+17}{(x+2)^3}\) using partial fractions.
  2. Expand \(\frac{-y^2+2y-4}{(y^2+4)^2}\) using partial fractions.
  3. Expand \(\frac{3r^3+r^2+3}{r^4+3r^2}\) using partial fractions.
  4. Find \(\int\frac{3z+2}{z^2+2z+1}\,dz\).
  5. Find \(\int\frac{3x^2+35}{x^3+5x}\,dx\).
  6. Find \(\int\frac{2v^3+4v^2+4v+2}{v^2+2v}\,dv\).
  7. Find \(\int\frac{2x^3+6x^2+4x+2}{(x+1)^2(x^2+1)}\,dx\).
  8. Describe the expansion of \(\frac{f(t)}{(t+1)^2(t^2+9)}\) using partial fractions. (Assume \(f(t)\) is a polynomial of degree less than 4.)
  9. Find \(\int \frac{-x^2+6x-3}{(x+3)(x^2+1)}\,dx\).

Solutions 1-4

Solutions 5-9

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 8.4