Question - Integration of Polynomial Function
mathintegration of polynomial functionintegral solvingsstep by step integrationantiderivative function
Solution:
To solve the integral provided in the image, we will integrate each term separately. The integral in question is:∫(8x^3 - x^2 + 5x - 1)dxEach term in the integral can be integrated by increasing the exponent by one and dividing by the new exponent. Don't forget to add the constant of integration at the end, which is typically denoted as 'C'.Here's how it is done step by step:∫8x^3 dx = 8 * ∫x^3 dx = 8 * (x^(3+1) / (3+1)) = 8 * (x^4 / 4) = 2x^4∫(-x^2) dx = -1 * ∫x^2 dx = -1 * (x^(2+1) / (2+1)) = -1 * (x^3 / 3) = -x^3 / 3∫5x dx = 5 * ∫x dx = 5 * (x^(1+1) / (1+1)) = 5 * (x^2 / 2) = (5/2) * x^2∫(-1) dx = -xNow, summing up all parts and adding the constant of integration, C, we get:2x^4 - (x^3 / 3) + (5/2) * x^2 - x + CSo, the antiderivative of the function 8x^3 - x^2 + 5x - 1 is:2x^4 - (x^3 / 3) + (5/2) * x^2 - x + C
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