Final answer:
To evaluate the integral ∫(3x² * a² - x²) dx from 0, split the integral into two parts and apply the power rule of integration to each term separately. Evaluate the integral at the upper and lower limits of integration. The value of the integral is 8a² - (8/3).
Step-by-step explanation:
To evaluate the integral ∫(3x² * a² - x²) dx from 0, we can use the power rule of integration. The integral of x^n, where n is a constant, is (x^(n+1))/(n+1). Using this rule, we can split the integral into two parts and apply the power rule to each term separately:
∫(3x² * a²) dx - ∫(x²) dx
By applying the power rule, we get:
(3/3)(x^3 * a²) - (1/3)(x^3)
Simplifying further:
x^3 * a² - (1/3)(x^3)
Now, we can evaluate the integral from 0:
Substituting the upper and lower limits of integration into the expression:
(2^3 * a²) - (1/3)(2^3) - (0^3 * a²) + (1/3)(0^3)
Simplifying:
8a² - (8/3)
So, the value of the integral from 0 is 8a² - (8/3).