Question

Use the indicated formula from the table of integrals in this section to find the indefinite integral. $$ \int \frac{4}{x^{2}-9} d x, \text { Formula } 29 $$

Short answer

The integral \( \int \frac{4}{x^{2}-9} d x \) equals \( \frac{2}{3} ln|\frac{x - 3}{x + 3}| + C \)

Step-by-step solution

Identify the Integral Form

The given integral is \( \int \frac{4}{x^{2}-9} d x \). This falls into the integral form of \( \int \frac{A}{n^{2}-a^{2}} d n \) as given in formula 29. Here, A=4 and a=3 (since \((-3)^2\) equals 9).

Apply the formula

In formula 29, it is stated that \( \int \frac{A}{n^{2}-a^{2}} d n = \frac{A}{2a}ln|\frac{n - a}{n + a}| + C \), where C is the constant of integration. Applying this formula gives \( \frac{4}{2*3} ln|\frac{x - 3}{x + 3}| + C \).

Simplify the expression

The expression simplifies to \( \frac{2}{3} ln|\frac{x - 3}{x + 3}| + C \).