Evaluating the Integral ∫(1/(64x^2 - 9)) dx

Integrating functions is a fundamental skill in calculus, and one that is often encountered in high school and college-level mathematics courses. In this article, we will walk through the process of evaluating the integral ∫(1/(64x^2 - 9)) dx, which involves a rational function.

Step 1: Identify the Integrand

The integrand in this case is the rational function 1/(64x^2 - 9). This means that the function we need to integrate is the reciprocal of the polynomial 64x^2 - 9.

Step 2: Choose an Appropriate Integration Technique

To evaluate this integral, we will use the method of partial fractions. This technique involves breaking down the rational function into a sum of simpler fractions, which can then be integrated individually.

Step 3: Decompose the Rational Function

The first step in the partial fractions method is to factor the denominator of the rational function, 64x^2 - 9. We can do this by finding the roots of the quadratic equation:

64x^2 - 9 = 0 x^2 - 9/64 = 0 x = ±√(9/64) = ±3/8

Now, we can write the rational function as a sum of two simpler fractions:

1/(64x^2 - 9) = A/(x - 3/8) + B/(x + 3/8)

Where A and B are constants that need to be determined.

Step 4: Determine the Values of A and B

To find the values of A and B, we can use the method of undetermined coefficients. This involves setting up a system of equations and solving for the unknown coefficients.

Substituting the values of x into the equation, we get:

1/(64(3/8)^2 - 9) = A/(3/8) + B/(-3/8) 1/(64(9/64) - 9) = A/(3/8) - B/(3/8)

Solving this system of equations, we find that A = 1/6 and B = -1/6.

Step 5: Integrate the Partial Fractions

Now that we have the decomposed form of the rational function, we can integrate each term separately:

∫(1/(64x^2 - 9)) dx = ∫(1/6 / (x - 3/8)) dx - ∫(1/6 / (x + 3/8)) dx

Applying the power rule of integration, we get:

∫(1/6 / (x - 3/8)) dx = (1/6) ln(x - 3/8) + C ∫(1/6 / (x + 3/8)) dx = -(1/6) ln(x + 3/8) + C

Combining these two results, we get the final answer:

∫(1/(64x^2 - 9)) dx = (1/6) ln(x - 3/8) - (1/6) ln(x + 3/8) + C

Where C is the constant of integration.

This step-by-step process demonstrates how to evaluate the integral ∫(1/(64x^2 - 9)) dx using the method of partial fractions. By breaking down the rational function and integrating the simpler fractions, we can arrive at the final result.