Answer: ln(1+sin²x) + C.
- A ln(1+sin²x) + C
- B 2 sin²x + C
- C tan⁻¹(sinx) + C
- D -cos 2x + C
Correct answer: A. ln(1+sin²x) + C
Explanation: Note sin2x = 2sinx cosx = d(sin²x)/dx. Substitution u=1+sin²x, du=sin2x dx. Integral = ∫du/u = ln|u|+C = ln(1+sin²x)+C. Since 1+sin²x≥1>0, absolute value is unnecessary.
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.