Final answer:
To find the equation for f(x) given f'(x), we can use integration. The equation is f(x) = (sinx)/2 + (3x/2) + 6.
Step-by-step explanation:
To find an equation for f(x) given that f'(x) = cos2x(1 + 3sinx) and f(0) = 6, we can use integration. The antiderivative of cos2x is (sinx)/2 + (x/2). Therefore, the equation for f(x) is:
f(x) = (sinx)/2 + (3x/2) + C
Using the initial condition f(0) = 6, we can solve for the constant C:
6 = (sin(0))/2 + (3(0))/2 + C
C = 6
So, the final equation for f(x) is:
f(x) = (sinx)/2 + (3x/2) + 6