Evaluate the integral of (4x-3)/(x^2-x-2)dx. a. (5/3)ln(x-2)+(7/3)ln(x+1)+c. b. ...

Question:

Partial Fractions:

The partial fraction decomposition is a technique that is often used to calculate integrals of rational functions: {eq}\displaystyle \int \dfrac{p(x)}{q(x)} \ dx {/eq}.

Here, {eq} p(x), \ q(x) {/eq} are polynomials. Normally, without loss of generality we can assume that {eq}\text{ deg } p(x) < \text{ deg } q(x) {/eq}. Otherwise we perform the long division and we can put it in that form where the numerator has degree less than the degree of the denominator.

In its simplest form it looks like:

{eq}\displaystyle \int \dfrac{p(x)}{(x-a)(x-b)} \ dx , \ \text{ deg } p(x) < 2 {/eq}.

In this case the rational function can be decomposed as:

$$\dfrac{p(x)}{(x-a)(x-b)} = \dfrac{A}{x - a} + \dfrac{B}{x - b} $$

where, A and B are to be determined by clearing out the denominators in the equation above and use that the equality between two polynomials can hold only when the coefficients of each power from each side are equal.

Then we can do the integral using the basic formula:

$$\displaystyle \int \dfrac{A}{x-a} dx = A \ln|x-a| + C $$

Note the absolute value in the result is necessary. The absolute value can be dropped by assuming {eq}x - a > 0 {/eq}.

Answer and Explanation: 1