Section 7.4: Integration by Partial Fractions (PFD)

Write out the form of the partial fraction decomposition. 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 following integrals:

1. \( \displaystyle{ \int \frac{x^2-2x}{x-1} \ dx}\) Solution ↗
2. \( \displaystyle{ \int \frac{1}{x^2-4} \ dx }\) Solution ↗
3. \( \displaystyle{ \int \frac{x}{x^2-x-2} \ dx }\) Solution ↗
4. \( \displaystyle{ \int \frac{2x+5}{x^2+3x+2} \ dx} \) Solution ↗
5. \( \displaystyle{\int \frac{x+2}{x^3-2x^2} \ dx }\) Solution ↗
6. \( \displaystyle{\int \frac{x^2+1}{(x-2)(x-5)^2} \ dx }\) Solution ↗
7. \( \displaystyle{\int \frac{2x}{(x+1)(x^2+1)} \ dx} \) Solution ↗
8. \( \displaystyle{\int_1^2 \frac{4y^2-7y-12}{y(y+2)(y-3)} \ dy }\) Solution ↗
9. \(\displaystyle{ \int_3^4 \frac{2x^2-4}{x^3-2x^2} \ dx}\) Solution ↗
10. \(\displaystyle{ \int \frac{x^2-x+6}{x^3+3x} \ dx}\) Solution ↗
11. \( \displaystyle{\int \frac{6x^3-3x^2-4x+7}{2x^2-x-1} \ dx }\) Solution ↗
12. \( \displaystyle{\int \frac{65x+7}{(8x+1)(x-1)} \ dx }\) Hints (PDF) ↗
13. \( \displaystyle{\int_0^1 \frac{2}{2x^2+3x+1} \ dx }\) Hints (PDF) ↗
14. \( \displaystyle{\int \frac{3x^2-10x+15}{(x-1)(x^2+2x+5)} \ dx }\) Solution ↗
15. \( \displaystyle{\int \frac{x^2-3x-3}{(x-1)^2(x^2+2x+2)} \ dx }\) Solution ↗