AP

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

Problem

Let

I(b)=\displaystyle \int_{0}^{b} x^{3} e^{2x}\,dx\,

for

b>0

. (a) Calculate

I(b)

and express your result as

e^{2b}

multiplied by a cubic polynomial in

b

minus a constant. (b) Verify your result in part (a) by differentiating your antiderivative and simplifying to recover the integrand

x^{3}e^{2x}

. (c) Use the structure of your answer in part (a) to derive, by inspection, a closed–form expression for

\displaystyle \int x^{n}e^{2x}\,dx

when

n

is a non-negative integer. Write your answer as

e^{2x}

multiplied by a finite sum written with sigma notation. (d) Apply the formula you obtained in part (c) to evaluate

\displaystyle \int_{0}^{\ln 2} x^{4} e^{2x}\,dx.