Answer: (1/3)ln|1+x³| + C.
- A (1/3)ln|1+x³| + C
- B ln|1+x³| + C
- C 1/(1+x³) + C
- D 3/(1+x³)² + C
Correct answer: A. (1/3)ln|1+x³| + C
Explanation: Substitution u = 1+x³, so du = 3x² dx, i.e. x² dx = du/3. Integral becomes ∫(1/u)(du/3) = (1/3)∫du/u = (1/3)ln|u| + C = (1/3)ln|1+x³| + 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.