Section 7.4: Integraion by Partial Fractions.

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

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

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

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