Question - Polynomial Integral by Term
Solution:
The integral presented in the image is of a polynomial. Integrating polynomials term by term, we find:∫(8x^3 - x^2 + 5x - 1) dxWe will integrate each term separately using the power rule for integrals. For a general term ax^n, the integral is (a/(n+1))x^(n+1), plus a constant of integration which we'll add at the end. Applying this rule to each term:∫8x^3 dx = (8/4)x^(3+1) = 2x^4∫-x^2 dx = (-1/3)x^(2+1) = -1/3 x^3∫5x dx = (5/2)x^(1+1) = 5/2 x^2∫-1 dx = -xAdding these results together and including the constant of integration C, we get the antiderivative:2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
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