Mathematics

Problem

For a nonzero constant

a

, define

I(a)=\displaystyle\int e^{ax}\sin(ax)\,dx

. a. Evaluate

I(a)

and express the antiderivative solely in terms of

a

x

, elementary functions, and a constant of integration. b. Rewrite your result from part (a) in the form

e^{ax}\bigl(A\sin(ax)+B\cos(ax)\bigr)+C

. Determine

A

and

B

explicitly in terms of

a

. c. Compute

\displaystyle\lim_{a\to 0}\frac{I(a)}{a}

, justifying each step of your limiting process, and interpret this limit in relation to the integral

\int x\,dx

. Explain the significance of your result.