Answer: -cos x + cos³x/3 + C.
- A -cos x + cos³x/3 + C
- B cos³x/3 + C, missing the linear cosine term
- C -cosx + C, missing the cubic cosine correction
- D sin⁴x/4 + C, an incorrect power-rule shortcut
Correct answer: A. -cos x + cos³x/3 + C
Explanation: Write sin³x = sinx·(1−cos²x). Substitute u=cosx, du=−sinx dx. Integral = −∫(1−u²)du = −u+u³/3+C. Back-substituting: −cosx+cos³x/3+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.