2.5 Integration of rational functions by partial fractions

Integration of rational functions by partial fractions

Lessons

Notes:
NOTE: 4 cases of partial fraction decomposition:

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

i.e.
7x(4x+1)(3x5)=A4x+1+B3x5 \frac{7x}{(4x+1)(3x-5)}=\frac{A}{4x+1}+\frac{B}{3x-5}
5x(x29)=5x(x+3)(x3)=Ax+Bx+3+Cx3 \frac{5}{x(x^2-9)}=\frac{5}{x(x+3)(x-3)}=\frac{A}{x}+\frac{B}{x+3}+\frac{C}{x-3}

CASE 2: Denominator is a product of linear factors with repeats.

i.e.
x6(4x+1)(3x5)2=A4x+1+B3x5+C(3x5)2\frac{x-6}{(4x+1)(3x-5)^2}=\frac{A}{4x+1}+\frac{B}{3x-5}+\frac{C}{(3x-5)^2}

1x2(7x4)(x+5)3=Ax+Bx2+C7x4+Dx+5+E(x+5)2+F(x+5)3 \frac{1}{x^2(7x-4)(x+5)^3}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{7x-4}+\frac{D}{x+5}+\frac{E}{(x+5)^2}+\frac{F}{(x+5)^3}

CASE 3: Denominator contains irreducible quadratic factors with no repeats.

i.e.
8x2(x3)(x2+x+1)=Ax3+Bx+CX2+x+1 \frac{8x^2}{(x-3)(x^2+x+1)}=\frac{A}{x-3}+\frac{Bx+C}{X^2+x+1}

5x(x2+9)=Ax+Bx+Cx2+9 \frac{5}{x(x^2+9)}=\frac{A}{x}+\frac{Bx+C}{x^2+9}

CASE 4: Denominator contains irreducible quadratic factors with repeats.

i.e.
5x2(x3)(x2+x+1)2=Ax3+Bx+Cx2+x+1+Dx+E(x2+x+1)2 \frac{5x^2}{(x-3)(x^2+x+1)^2}=\frac{A}{x-3}+\frac{Bx+C}{x^2+x+1}+\frac{Dx+E}{(x^2+x+1)^2}

7+x8(x28)(x2+25)3=1+x10(x2)(x2+2x+4)(x2+25)3=Ax2+Bx+Cx2+2x+4+Dx+Ex2+25+Fx+G(x2+25)2+Hx+1(x2+25)3 \frac{7+x^8}{(x^2-8)(x^2+25)^3}=\frac{1+x^{10}}{(x-2)(x^2+2x+4)(x^2+25)^3}=\frac{A}{x-2}+\frac{Bx+C}{x^2+2x+4}+\frac{Dx+E}{x^2+25}+\frac{Fx+G}{(x^2+25)^2}+\frac{Hx+1}{(x^2+25)^3}
  • 1.
    CASE 1: Denominator is a product of linear factors with no repeats.

    Evaluate the integral.
  • 2.
    CASE 2: Denominator is a product of linear factors with repeats.

    Evaluate the integral.
    • b)
      18xx39x2+15x+25dx\int \frac{18x}{x^3-9x^2+15x+25}dx
    • c)
      dxx2(x1)2\int \frac{dx}{x^2(x-1)^2}
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Integration of rational functions by partial fractions

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