Integral of a Polynomial Expression
Certainly! The given expression is an integral that you need to evaluate. You have the following expression:
∫(8x^3 - x^2 + 5x - 1) dx
To solve this, you integrate each term separately with respect to x. The integral of a sum or difference of functions is the sum or difference of their integrals. Here's how to integrate each term:
1. ∫8x^3 dx = (8/4)x^(3+1) = 2x^4
2. ∫-x^2 dx = -(1/3)x^(2+1) = -(1/3)x^3
3. ∫5x dx = (5/2)x^(1+1) = (5/2)x^2
4. ∫-1 dx = -x
Now combine these results to get the complete antiderivative:
∫(8x^3 - x^2 + 5x - 1) dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
Here, C represents the constant of integration, which is a standard addition in indefinite integrals, as there are an infinite number of antiderivatives differing by a constant.
