1. What “form” should the partial-fraction decomposition of each of these rational expressions have? I.e. just write a template, don’t find the numerators explicitly. \[ \frac{1}{(x-1)^2(x+1)^2\left(x^2-1\right)\left(x^2+1\right)} \qquad \frac{1}{\left(x^2+2\right)^2\left(x^2-x\right)\left(x^2+2x+1\right)} \]
2. Demonstrate how to evaluate each of the following integrals. \[ \int \frac{5x^2-13x+12}{x-2} \,\mathrm{d}x \qquad \int \frac{5x+6}{x^2-2x} \,\mathrm{d}x \qquad \int \frac{x^2+x+27}{x^3+9x} \,\mathrm{d}x \] \[ \int \frac{x^3+1}{x^3+x^2} \,\mathrm{d}x \qquad \int \frac{\mathrm{d}x}{x^2+9} \qquad \int \frac{x^2+3x-9}{(x+9)(x^2+9)} \,\mathrm{d}x \] \[ \int \frac{7x^2-7x+2}{x^3-x^2-x+1} \,\mathrm{d}x \qquad \int \frac{4x^3-23x^2+44x-31}{x^4-6x^3+14x^2-14x+5} \,\mathrm{d}x \]