Question - Integrating Polynomials Using Power Rule
Solution:
To solve the integral provided in the image:∫(8x^3 - x^2 + 5x - 1)dxYou need to integrate each term separately using the power rule for integration. The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1), where n ≠ -1.Applying the power rule to each term:For 8x^3,∫8x^3 dx = 8 * ∫x^3 dx = 8 * (x^(3+1))/(3+1) = 8 * (x^4)/4 = 2x^4For -x^2,∫(-x^2) dx = - ∫x^2 dx = - (x^(2+1))/(2+1) = - (x^3)/3For 5x,∫5x dx = 5 * ∫x dx = 5 * (x^(1+1))/(1+1) = 5 * (x^2)/2 = (5/2)x^2For -1,∫(-1) dx = - ∫1 dx = -xNow, combine all the integrated terms:∫(8x^3 - x^2 + 5x - 1)dx = 2x^4 - (x^3)/3 + (5/2)x^2 - x + CWhere C is the constant of integration.
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