Question - Indefinite Integral of a Polynomial
Solution:
To solve the given indefinite integral, integrate each term separately with respect to x.The integral you provided is:∫(8x^3 - x^2 + 5x - 1) dxApply the power rule of integration (which states that ∫x^n dx = (x^(n+1)) / (n+1) + C for any n that is not equal to -1) to each term:∫8x^3 dx = (8/4)x^(3+1) + C_1 = 2x^4 + C_1∫-x^2 dx = (-1/3)x^(2+1) + C_2 = -(1/3)x^3 + C_2∫5x dx = (5/2)x^(1+1) + C_3 = (5/2)x^2 + C_3∫-1 dx = -x + C_4Combine these integrals and their constants into one expression:2x^4 - (1/3)x^3 + (5/2)x^2 - x + Cwhere C is the constant of integration that combines C_1, C_2, C_3, and C_4 into a single constant, since any finite number of constants can be combined into one. So the integral of the expression is:2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
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