Answer: ex(sinx + cosx)/2 + C.
- A ex(sinx + cosx)/2 + C
- B ex cosx + C
- C ex sinx + C
- D ex(cosx - sinx)/2 + C
Correct answer: A. ex(sinx + cosx)/2 + C
Explanation: Integration by parts twice: I=∫eˣcosx dx. First: u=cosx, dv=eˣdx → I=eˣcosx+∫eˣsinx dx. Second: ∫eˣsinx dx=eˣsinx−I. So 2I=eˣ(cosx+sinx), giving I=eˣ(sinx+cosx)/2+C.
The exact area under the curve from a to b (shaded) is approximated by a few rectangles, the Riemann sum idea behind the definite integral.