Solving Integral of a Polynomial Function
The image shows an integral that you'd like to solve:
\[
\int (8x^3 - x^2 + 5x - 1) \, dx
\]
To solve this integral, integrate each term individually with respect to \( x \). When integrating a polynomial term of the form \( x^n \), the antiderivative is \( \frac{x^{n+1}}{n+1} \) provided that \( n \neq -1 \).
\[
\int 8x^3 \, dx = \frac{8}{4}x^{4} = 2x^4
\]
\[
\int (-x^2) \, dx = -\frac{1}{3}x^{3}
\]
\[
\int 5x \, dx = \frac{5}{2}x^{2}
\]
\[
\int (-1) \, dx = -x
\]
Now summing all of the individual antiderivatives and adding a constant of integration \( C \), we have:
\[
2x^4 - \frac{1}{3}x^{3} + \frac{5}{2}x^{2} - x + C
\]
So the integral of \( 8x^3 - x^2 + 5x - 1 \) with respect to \( x \) is:
\[
\int (8x^3 - x^2 + 5x - 1) \, dx = 2x^4 - \frac{1}{3}x^{3} + \frac{5}{2}x^{2} - x + C
\]
