Section 7.4: Integraion by Partial Fractions. (Partial Fraction Decomposition, PFD)


Write out the form of the partial fraction decomposition of the function. Do not determine the numerical values of the coefficients.

I. (a) \( \displaystyle{ \frac{x^2}{x^2+x+12}} \)   (b) \( \displaystyle{ \frac{x-2}{x^2-x-12}} \)   solution

II. (a) \(\displaystyle{ \frac{x^4+2}{x^5+3x^2}} \)   (b) \( \displaystyle{\frac{2}{(x^2-4)^2}} \)   solution

Evaluate the integrals:

  1. \( \displaystyle{ \int \frac{x^2-2x}{x-1} \ \mathrm{d}x}\) solution

  2. \( \displaystyle{ \int \frac{1}{x^2-4} \ \mathrm{d}x }\) solution

  3. \( \displaystyle{ \int \frac{x}{x^2-x-2} \ \mathrm{d}x }\) solution

  4. \( \displaystyle{ \int \frac{2x+5}{x^2+3x+2} \ \mathrm{d}x} \) solution

  5. \( \displaystyle{\int \frac{x+2}{x^3-2x^2} \ \mathrm{d}x }\) solution

  6. \( \displaystyle{\int \frac{x^2+1}{(x-2)(x-5)^2} \ \mathrm{d}x }\) solution

  7. \( \displaystyle{\int \frac{2x}{(x+1)(x^2+1)} \ \mathrm{d}x} \) solution

  8. \( \displaystyle{\int_1^2 \frac{4y^2-7y-12}{y(y+2)(y-3)} \ \mathrm{d}y }\) solution

  9. \(\displaystyle{ \int_3^4 \frac{2x^2-4}{x^3-2x^2} \ \mathrm{d}x}\) solution

  10. \(\displaystyle{ \int \frac{x^2-x+6}{x^3+3x} \ \mathrm{d}x}\) solution

  11. \( \displaystyle{\int \frac{6x^3-3x^2-4x+7}{2x^2-x-1} \ \mathrm{d}x }\) solution

  12. \( \displaystyle{\int \frac{65x+7}{(8x+1)(x-1)} \ \mathrm{d}x }\) hints

  13. \( \displaystyle{\int_0^1 \frac{2}{2x^2+3x+1} \ \mathrm{d}x }\) hints

  14. \( \displaystyle{\int \frac{3x^2-10x+15}{(x-1)(x^2+2x+5)} \ \mathrm{d}x }\) solution

  15. \( \displaystyle{\int \frac{x^2-3x-3}{(x-1)^2(x^2+2x+2)} \ \mathrm{d}x }\) solution