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

1. Sep 20, 2005

amcavoy

I already know the answer to this, but would like your opinion on the quickest way to calculate integrals like these. I am finding myself evaluating similar integrals in my diffeq homework, and the method that I used (parts and partial fractions) is a mess and takes way too long (if it were to appear on a test).

$$\int\left(\frac{1-x}{x^2-4}\right)dx$$

Thanks.

2. Sep 20, 2005

LeonhardEuler

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:
$$\int-\frac{x-1}{x^2-4}dx$$
$$=-\int\frac{x-2}{(x+2)(x-2)}+\frac{2}{2(x+2)(x-2)}dx$$
$$=-\int\frac{1}{x+2}+\frac{2+x}{2(x+2)(x-2)}-\frac{x}{2(x^2-4)}}dx$$
$$=-\int\frac{1}{x+2}+\frac{1}{2(x-2)}-\frac{x}{2(x^2-4)}}dx$$
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

3. Sep 20, 2005

amcavoy

Well it looks like the only way to save time here is to use my TI-89 .

Thanks for the reply Euler.

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