The integral of sec^3(4x) can be found using the substitution method.
Let u = 4x, then du/dx = 4 and dx = du/4
Substitute these values into the integral:
- ∫ sec^3(4x) dx = ∫ sec^3(u) (du/4)
Now we can use the formula for the integral of sec^3(x):
- ∫ sec^3(x) dx = (1/2) sec(x) tan(x) + (1/2) ln |sec(x) + tan(x)| + C
Substitute back in u for x:
- ∫ sec^3(4x) dx = (1/8) sec(4x) tan(4x) + (1/8) ln |sec(4x) + tan(4x)| + C
Therefore, the integral of sec^3(4x) is (1/8) sec(4x) tan(4x) + (1/8) ln |sec(4x) + tan(4x)| + C, where C is the constant of integration.
~ Zeph