Final answer:
The correct substitution to simplify the integral of a sec²(\theta) / a²(1 + tan²(\theta)) is c) Set x = a tan(\theta), which takes advantage of the derivative of tan(\theta) being sec²(\theta) and the Pythagorean identity related to secant and tangent.
Step-by-step explanation:
To solve the integral of a sec²(\theta) / a²(1 + tan²(\theta)), the most useful substitution would be to stabilize the relationship between secant and tangent, as they are related through the Pythagorean identity sec²(\theta) = 1 + tan²(\theta). Recognizing that derivative of tan(\theta) is sec²(\theta), the most effective substitution would be c) Set x = a tan(\theta). This substitution directly leverages the differential of tangent and simplifies the integral greatly.
Considering the integral's form and the Pythagorean identities, using x = a tan(\theta) allows us to both simplify the sec²(\theta) term and deal with the a² appearing in the denominator. Upon substituting, the integral becomes much easier to solve as the troublesome secant squared term is appropriately dealt with. Always when performing substitutions, it is critical to also change differential elements, so d\theta would also be substituted with an expression in terms of dx.