Question

(c) Solve: \[ \int \frac{3x + 5}{x^3 - x^2 - x + 1} \, dx. \]

Hint:

Partial fractions simplify the integration of rational functions.

Answer 1

Factorize the denominator: \[ x^3 - x^2 - x + 1 = (x - 1)(x^2 + 1). \] Use partial fractions: \[ \frac{3x + 5}{x^3 - x^2 - x + 1} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 + 1}. \] Find \( A, B, \) and \( C \), then integrate: \[ \int \frac{A}{x - 1} \, dx + \int \frac{Bx + C}{x^2 + 1} \, dx. \]