Question

Use partial fractions to find the indefinite integral. $$ \int \frac{1}{4 x^{2}-9} d x $$

Short answer

The indefinite integral of \( \frac{1}{4x^{2}-9} \) is \( \frac{1}{5} \ln |2x - 3| - \frac{1}{5} \ln |2x + 3| + C \)

Step-by-step solution

Factor the Denominator

Factoring the denominator yields: \( 4x^{2}-9 = 2x - 3)(2x + 3) \)

Set Up the Partial Fraction Decomposition

\( \frac{1}{4x^{2}-9} = \frac{A}{2x - 3} + \frac{B}{2x + 3} \). Our task is to find the parameters A and B.

Solve for Parameters A and B

Multiply both sides by \(4x^{2}-9=(2x - 3)(2x + 3)\) to clear the fraction. This yields: \(1 = A(2x + 3) + B(2x - 3)\). Solve this equation for A and B by choosing smart x-values where A or B cancel. If you choose \(x=\frac{3}{2}\), you get \(1=5A \Rightarrow A=\frac{1}{5}\). If you choose \(x=-\frac{3}{2}\), you get \(1=-5B \Rightarrow B=-\frac{1}{5}\). So \( \frac{1}{4x^{2}-9} = \frac{\frac{1}{5}}{2x - 3} - \frac{\frac{1}{5}}{2x + 3} \)

Integrate Each Term

The integral of a sum is the sum of the integrals, allowing each term to be integrated separately. The integral of each term is a natural logarithm, yielding the final answer: \( \int \frac{1}{4 x^{2}-9} dx = \frac{1}{5} \ln |2x - 3| - \frac{1}{5} \ln |2x + 3| + C \)