Integration of rational functions by partial fractions

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Examples
Lessons
  1. CASE 1: Denominator is a product of linear factors with no repeats.

    Evaluate the integral.
    1. 10x2+20x10x(x+2)(2x1)dx \int \frac{10x^2+20x-10}{x(x+2)(2x-1)}dx
    2. 18x39xdx\int \frac{18}{x^3-9x}dx
  2. CASE 2: Denominator is a product of linear factors with repeats.

    Evaluate the integral.
    1. x24(x1)3dx \int \frac{x^2-4}{(x-1)^3}dx
    2. 18xx39x2+15x+25dx\int \frac{18x}{x^3-9x^2+15x+25}dx
  3. CASE 3: Denominator contains irreducible quadratic factors with no repeats.

    *** recall: dxx2+a2=1atan1(xa)+C\int \frac{dx}{x^2+a^2}=\frac{1}{a}tan^{-1}(\frac{x}{a})+C ***


    Evaluate the integral.
    1. 3x2x+9x3+9xdx\int \frac{3x^2-x+9}{x^3+9x}dx
  4. CASE 4: Denominator contains irreducible quadratic factors with repeats.

    Evaluate 3x3+3x23x+2x(x2+1)2dx \int \frac{-3x^3+3x^2-3x+2}{x(x^2+1)^2}dx
    1. First perform long division, then partial fraction decomposition

      Evaluate the integral.
      1. 3x4+13x3+22x2100x10x33x210xdx \int \frac{-3x^4+13x^3+22x^2-100x-10}{x^3-3x^2-10x}dx