Final answer:
To evaluate the integral ∫(4x² - 8x³) dx from 1 to 4, use the power rule of integration to find the antiderivative. Then substitute the limits of integration into the antiderivative and simplify the expression.
Step-by-step explanation:
To evaluate the given integral, we can use the power rule of integration. The power rule states that if the integral of x^n with respect to x equals (1/(n+1))x^(n+1) + C, where C is the constant of integration. Applying this rule to the given integral, we have:
∫(4x² - 8x³) dx = (4/3)x^3 - (8/4)x^4 + C
Next, we can evaluate the definite integral by substituting the limits of integration, 1 and 4, into the antiderivative.
∫(4x² - 8x³) dx from 1 to 4 = (4/3)(4^3) - (8/4)(4^4) - [(4/3)(1^3) - (8/4)(1^4)]
Simplifying further, we get:
(4/3)(64) - (2)(256) - [(4/3)(1) - (2)] = 256/3 - 512 - (4/3) + 2 = -488/3