Exercise 6

For each of the following integrals involving rational functions, (1) use a CAS to find the partial fraction decomposition of the integrand; (2) evaluate the integral of the resulting function without the assistance of technology; (3) use a CAS to evaluate the original integral to test and compare your result in (2).

1. $\int \frac{x^3 + x + 1}{x^4 - 1} \, dx$

2. $\int \frac{x^5 + x^2 + 3}{x^3 - 6x^2 + 11x - 6} \, dx$

3. $\int \frac{x^2 - x - 1}{(x-3)^3} \, dx$