Question

Find \( \int \frac{3x + 1}{(x - 2)^2 (x + 2)} \, dx \)

Hint:

When dealing with rational functions, use partial fraction decomposition to split the integrand into simpler terms that are easier to integrate.

Answer 1

We are given the integral: \[ I = \int \frac{3x + 1}{(x - 2)^2 (x + 2)} \, dx \] This is a rational function, and to solve it, we can use partial fraction decomposition. First, express the integrand as a sum of partial fractions: \[ \frac{3x + 1}{(x - 2)^2 (x + 2)} = \frac{A}{x - 2} + \frac{B}{(x - 2)^2} + \frac{C}{x + 2} \] Multiply both sides by \( (x - 2)^2 (x + 2) \) to clear the denominators: \[ 3x + 1 = A(x - 2)(x + 2) + B(x + 2) + C(x - 2)^2 \] Now, expand and solve for \( A \), \( B \), and \( C \) by equating the coefficients of powers of \( x \). Once the partial fractions are determined, integrate each term separately.