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