Activity 5.5.2

For each of the following problems, evaluate the integral by using the partial fraction decomposition provided.

1. $\displaystyle \int \frac{1}{x^2 - 2x - 3} \, dx$, given that $\displaystyle \frac{1}{x^2 - 2x - 3} = \frac{1/4}{x-3} - \frac{1/4}{x+1}$

2. $\displaystyle \int \frac{x^2+1}{x^3 - x^2} \, dx$, given that $\displaystyle \frac{x^2+1}{x^3 - x^2} = -\frac{1}{x} - \frac{1}{x^2} + \frac{2}{x-1}$

3. $\displaystyle \int \frac{x-2}{x^4 + x^2}\, dx$, given that $\displaystyle \frac{x-2}{x^4 + x^2} = \frac{1}{x} - \frac{2}{x^2} + \frac{-x+2}{1+x^2}$