Answer: Recursion: I n = -(sin n-1 x cosx)/n + (n-1)/n × I_(n-2).
- A Integrating by parts repeatedly until the power is gradually reduced to zero
- B A single substitution u = sin x with no recursive structure
- C Recursion: I<sub>n</sub> = -(sin<sup>n-1</sup>x cosx)/n + (n-1)/n × I_(n-2)
- D A direct closed-form formula equal to n × ln(sinx)
Correct answer: C. Recursion: I<sub>n</sub> = -(sin<sup>n-1</sup>x cosx)/n + (n-1)/n × I_(n-2)
Explanation: Integration by parts: u=sinⁿ⁻¹x, dv=sinx dx. After differentiating and substituting cos²x=1−sin²x, the recursion Iₙ = −sinⁿ⁻¹x cosx/n + (n−1)/n·Iₙ₋₂ emerges, reducing the power by 2 each step.
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.