Final answer:
Using substitution u=4x transforms the integral ∠ 2^6 f(4x)dx into one with known limits from 8 to 24, making it ∠ 8^24 f(u) ⅔ du and resulting in an answer of ¼.
Step-by-step explanation:
To calculate the integral ∠ 2^6 f(4x)dx, given that 1∠ 8^24 f(x)dx = 1, a useful approach is to use the substitution u = 4x. When we apply this substitution, both the differential dx and the limits of integration will change accordingly. First, we differentiate both sides of the substitution with respect to x to get du = 4dx, or dx = ⅔ du. After changing the differential, we also need to adjust the limits of integration. Because u = 4x, when x=2, u=8, and when x=6, u=24. The integral now becomes ∠ 8^24 f(u) ⅔ du.
We are given that 1∠ 8^24 f(x)dx = 1. Since the limits of integration 8 to 24 match and because we introduced a factor of 1/4 when changing the differential, the evaluated integral would now be 1/4 of the given integral value. Therefore, we end up with ∠ 2^6 f(4x)dx = ¼.