Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Use partial fractions to find the indefinite integral. $$ \int \frac{5}{x^{2}+x-6} d x $$

Step-by-Step Solution

Answer

The indefinite integral of the given fraction is \(ln | \frac{x - 2}{x + 3} | + C\)

Step 1: Factor the Denominator

The denominator of the fraction \(x^{2} + x - 6\) can be factored to \((x-2)(x+3)\). This gives us \(\int \frac{5}{(x-2)(x+3)} dx\)

Step 2: Decompose into Partial Fractions

Decompose the fraction into the form \(\frac{A}{x-2} + \frac{B}{x+3}\). To find the values of A and B, balance the two fractions: \(5 = A(x+3) + B(x-2)\). With x=2, we find A=1. With x=-3, we find B=-1. So, our decomposition is \(\int ( \frac{1}{x-2} - \frac{1}{x+3} ) dx\)

Step 3: Integrate

We use standard formulas to integrate, remembering to put the integral sign in front of each fraction. The integral of \(\frac{1}{x-a}\) is \(ln \lvert x-a \rvert\). Thus our integral is \(ln \lvert x-2 \rvert - ln \lvert x+3 \rvert + C\), where C is our constant of integration. Apply the properties of logarithms to simplify to \(ln \lvert \frac{x-2}{x+3} \rvert + C\)