Answer: e x (sinx + cosx)/2 + C.
- A e<sup>x</sup>(sinx + cosx)/2 + C
- B e<sup>x</sup> cosx + C
- C e<sup>x</sup> sinx + C
- D e<sup>x</sup>(cosx - sinx)/2 + C
Correct answer: A. e<sup>x</sup>(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.