MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
A student is evaluating the integral ∫(x2+3x+2)5/22x+3dx. Which of the following statements best explains why rewriting x2+3x+2 in completed-square form is a more productive first step than factoring it?
(a) Completing the square changes the denominator into the form (x+23)2−41, which suggests an immediate trigonometric substitution and avoids any algebraic expansion.
(b) Factoring would force the use of partial fractions, but the method fails when the denominator is raised to the non-integer power 25.
(c) Completing the square produces an inner expression whose derivative is proportional to 2x+3, creating a direct path to an antiderivative.
(d) If you factor first and then raise the result to the 25 power, the denominator becomes a fourth-degree expression, making the integral far more complicated.