Question - Evaluating Definite Integrals of Polynomial Functions using Power Rule
Solution:
This is a definite integral of a polynomial function, which we can evaluate using the Power Rule for integration. The Power Rule states that the integral of x^n is (x^(n+1)) / (n+1) plus a constant (C), for any real number n ≠ -1.So let's evaluate your integral:∫(8x^3 - x^2 + 5x - 1) dxIntegration term by term:∫8x^3 dx = 8 * ∫x^3 dx = 8 * (x^(3+1) / (3+1)) = 2 * x^4∫-x^2 dx = - ∫x^2 dx = - (x^(2+1) / (2+1)) = - (1/3) * x^3∫5x dx = 5 * ∫x dx = 5 * (x^(1+1) / (1+1)) = (5/2) * x^2∫-1 dx = -xCombining these results:2x^4 - (1/3)x^3 + (5/2)x^2 - xWithout limits of integration given, we cannot evaluate for specific numbers. If there were limits, you would substitute the upper limit into the result for x, then subtract the result from substituting the lower limit. Since no limits are provided, this is the indefinite integral result, and don't forget to add the constant of integration C:2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
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