Integration of Polynomials
To solve the given integral, integrate each term with respect to x:
\[
\int (8x^3 - x^2 + 5x - 1) \, dx
\]
Integrate term by term:
\[
= \int 8x^3 \, dx - \int x^2 \, dx + \int 5x \, dx - \int 1 \, dx
\]
Now apply the power rule of integration which is:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{where} \quad n \neq -1
\]
So we get:
\[
= \frac{8x^{3+1}}{3+1} - \frac{x^{2+1}}{2+1} + \frac{5x^{1+1}}{1+1} - x + C
\]
Simplify the expression:
\[
= \frac{8x^4}{4} - \frac{x^3}{3} + \frac{5x^2}{2} - x + C
\]
\[
= 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C
\]
Here, \( C \) is the constant of integration.
So the final answer is:
\[
2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C
\]
