AP

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

Problem

Let

I=\displaystyle\int e^{3x}\sin(2x)\,dx

. (1) Evaluate

\displaystyle\int e^{3x}\cos(2x)\,dx

and express your result in terms of

I

and elementary functions. (2) Use your result from part (a) to derive an algebraic equation involving only

I

and elementary functions. Solve this equation to obtain an explicit antiderivative of

e^{3x}\sin(2x)

. (3) Differentiate the expression you found in part (b) to confirm that its derivative equals

e^{3x}\sin(2x)

. Explain each step of your verification.