Questions: Evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) ∫ 3/(x-1)(x+2) dx

Solution Steps

To evaluate the integral \(\int \frac{3}{(x-1)(x+2)} dx\), we can use the method of partial fraction decomposition. This involves expressing the integrand as a sum of simpler fractions, which can then be integrated individually. Specifically, we express \(\frac{3}{(x-1)(x+2)}\) as \(\frac{A}{x-1} + \frac{B}{x+2}\) and solve for \(A\) and \(B\). Once we have these constants, we integrate each term separately and include the constant of integration \(C\).

We start with the integral

\[ \int \frac{3}{(x-1)(x+2)} dx. \]

Using partial fraction decomposition, we express the integrand as

\[ \frac{3}{(x-1)(x+2)} = \frac{-1}{x+2} + \frac{1}{x-1}. \]

Next, we integrate each term separately:

\[ \int \left( \frac{-1}{x+2} + \frac{1}{x-1} \right) dx = \int \frac{-1}{x+2} dx + \int \frac{1}{x-1} dx. \]

The integrals yield:

\[ -\log|x+2| + \log|x-1|. \]

Combining the results from the integration, we have:

\[ \log|x-1| - \log|x+2| + C, \]

where \(C\) is the constant of integration.

Final Answer