Polynomial Integral by Term
The integral presented in the image is of a polynomial. Integrating polynomials term by term, we find:
∫(8x^3 - x^2 + 5x - 1) dx
We 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 = -x
Adding 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
