AP

MATHEMATICS · Multi-part · AP · Calculus BC · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Let

F(x)=\displaystyle \int x^{5}\cos x\,dx

and let

G=\displaystyle \int_{0}^{\tfrac{\pi}{2}} x^{5}\cos x\,dx.

(a) Determine

F(x)

and write your result in the form

P(x)\sin x+Q(x)\cos x,

where

P

and

Q

are polynomials of degree at most 5. (b) Evaluate

G

exactly. (c) Extend your work to conjecture a general closed–form expression for

\displaystyle \int x^{2n+1}\cos x\,dx,

where

n

is a non-negative integer. State your conjecture clearly and justify the polynomial degree of each trigonometric coefficient.