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