Final answer:
The question asks for the most appropriate substitution to simplify an indefinite integral involving a square root of a binomial square. The best substitution is (a) \(u = 7x - 4\), which allows us to use trigonometric identity to simplify the integral.
Step-by-step explanation:
The student is trying to evaluate an indefinite integral with the form \(\int \frac{1}{\sqrt{1 - (7x-4)^2}} \, dx\). The most appropriate substitution to simplify this integral is, in fact, trigonometric substitution. We recognize the inner part of the square root as a form that suggests the Pythagorean trigonometric identity \(\sin^2(\theta) + \cos^2(\theta) = 1\). Therefore, the best substitution in this case would be (a) \(u = 7x - 4\), after which we can use the trigonometric identity \(\sin(\theta) = \frac{u}{a}\) where in this case, a is 1, for further simplification.
After the substitution, you can differentiate \(u\) to find \(du\) and replace \(dx\) accordingly in the integrand. The ultimate goal is to manipulate the integral into a form where the trigonometric identity can be used to simplify and evaluate the integral.