Answer: x√(1+x²)/2 + (1/2)sinh⁻¹x + C.
- A x√(1+x²)/2 + (1/2)sinh⁻¹x + C
- B x√(1+x²) + C, omitting the inverse hyperbolic sine term
- C √(1+x²) + C, treating the integral as if it had no x factor
- D x²/√(1+x²) + C, obtained from differentiating instead of integrating
Correct answer: A. x√(1+x²)/2 + (1/2)sinh⁻¹x + C
Explanation: Trig sub x = tanθ, dx = sec²θ dθ, √(1+x²) = secθ. ∫sec³θ dθ = (secθ tanθ)/2 + (1/2)ln|secθ+tanθ| + C. Back-substituting: x√(1+x²)/2 + (1/2)sinh⁻¹x + 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.