Final answer:
For the integral containing √(x²-81), the suggested change of variables is x = 9cosθ as it allows simplification using the trigonometric identity cos²θ + sin²θ = 1.
correct option is C. x = 9secθ
Step-by-step explanation:
To solve integrals containing the square root of x²-81, such as √(x²-81), a trigonometric substitution is often recommended to simplify the integrand. In this case, the appropriate substitution would attempt to match the expression within the square root to a trigonometric identity. Noting the identity for cosine, cos²θ = 1 - sin²θ, and rearranging to sin²θ = 1 - cos²θ, we can match the integrand's form if we let x = 9cosθ because then x² becomes 81cos²θ. This form suggests a trigonometric identity since cos²θ + sin²θ equals 1.
Therefore, the change of variable will yield a familiar trigonometric form: √(81cos²θ - 81) = 9√(1 - cos²θ) = 9sinθ, which simplifies the integral significantly.