Using the method of substitution, we have:
I = ∫(3 to 2) x^2(2 + 2x^3)^2 dx
Let u = 2 + 2x^3
Then, du = 6x^2 dx
To find the limits of integration:
When x = 3, u = 2 + 2(3)^3 = 56
When x = 2, u = 2 + 2(2)^3 = 18
Therefore, a = 18 and b = 56
Now, we need to express the integrand in terms of u:
f(u) = x^2(2 + 2x^3)^2
f(u) = (u - 2)^2
f(u) = u^2 - 4u + 4
So, the integral becomes:
I = ∫(18 to 56) (u^2 - 4u + 4) du