Homework Help: How to find integrals like int (1-x)(x^2-4)dx

In general, partial fractions is the only way to go. In this particular case, you can split the integral up like this to do it pretty fast:
[tex]\int-\frac{x-1}{x^2-4}dx[/tex]
[tex]=-\int\frac{x-2}{(x+2)(x-2)}+\frac{2}{2(x+2)(x-2)}dx[/tex]
[tex]=-\int\frac{1}{x+2}+\frac{2+x}{2(x+2)(x-2)}-\frac{x}{2(x^2-4)}}dx[/tex]
[tex]=-\int\frac{1}{x+2}+\frac{1}{2(x-2)}-\frac{x}{2(x^2-4)}}dx[/tex]
Now you can directly compute the first two parts as logs and the last as a log by recognizing x as one quarter the derivative of the denomenator