Step 1: Identify the integral you need to evaluate, which is ∫ (4x^3 - 2x) dx.
Step 2: Break the integral into two parts: ∫ 4x^3 dx and ∫ -2x dx.
Step 3: For the first part, ∫ 4x^3 dx, use the power rule of integration. The power rule states that ∫ x^n dx = (1/(n+1))x^(n+1) + C. Here, n = 3, so you get (4/(3+1))x^(3+1) = (4/4)x^4.
Step 4: For the second part, ∫ -2x dx, again use the power rule. Here, n = 1, so you get (-2/(1+1))x^(1+1) = (-2/2)x^2.
Step 5: Combine the results from Step 3 and Step 4. You have (4/4)x^4 - (2/2)x^2.
Step 6: Simplify the expression. (4/4)x^4 becomes x^4 and (-2/2)x^2 becomes -x^2.
Step 7: Add the constant of integration, C, to the final result. The final answer is x^4 - x^2 + C.
Integration – The process of finding the integral of a function, which represents the area under the curve of that function.
Power Rule for Integration – A rule that states the integral of x^n is (x^(n+1))/(n+1) + C, applicable for n ≠ -1.
