Antiderivative of a Polynomial Function
Certainly! The expression you provided is an integral, and we will find the antiderivative (indefinite integral) of the function. Here's the integral:
\[
\int (8x^3 - x^2 + 5x - 1) \, dx
\]
To solve this integral, we will integrate each term separately. Recall the power rule for integration, \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) where C is the constant of integration. Applying this rule to each term:
1. The integral of \(8x^3\) is \(\frac{8}{4}x^{4} = 2x^4\).
2. The integral of \(-x^2\) is \(-\frac{1}{3}x^{3} = -\frac{1}{3}x^3\).
3. The integral of \(5x\) is \(\frac{5}{2}x^{2} = \frac{5}{2}x^2\).
4. The integral of \(-1\) is \(-x\).
Now, putting it all together and adding the constant of integration, \(C\), the antiderivative is:
\[
\int (8x^3 - x^2 + 5x - 1) \, dx = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C
\]
