The best change of variables for an integral containing √x² + 144 is C) x = 12sec(θ).
Analysis:
Form of the integral: We are dealing with an integral containing √x² + a², where a² = 144. This suggests a trigonometric substitution involving secant or cosecant functions.
Relationship between secant and x²: The secant function (sec(θ)) is defined as the reciprocal of cos(θ): sec(θ) = 1/cos(θ). Squaring both sides, we get: sec²(θ) = 1/cos²(θ) = 1 + tan²(θ). Rewriting this as x² + 1 = sec²(θ), we see that the form matches the integral's argument (x² + 144).
Therefore, substituting x = 12sec(θ) will transform the integral into a form where it is easier to solve.
Here's a breakdown of the other options:
A) x=12sin(θ): This substitution doesn't match the form of the integral.
B) x=12cos(θ): While cos²θ is related to x², the substitution still doesn't fully match the integral's argument.
D) x=12csc(θ): Similar to option A, this doesn't directly relate to the integral's form.