Evaluate the integral J= x^ 2 +2x+1 x^ 2 +2x+5 ,dx
MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
Evaluate the integral
J=∫x2+2x+5x2+2x+1dx,
and use it to answer the following questions.
(1) Rewrite the integrand as the sum of a constant and a rational function whose numerator is linear. Show each algebraic step leading to this decomposition.
(2) Using the form obtained in part (a), determine an explicit antiderivative for J. Simplify your result as much as possible.
(3) Apply your antiderivative to compute ∫−12x2+2x+5x2+2x+1dx. Present the exact value and outline the reasoning you used to reach it.