find the most general antiderivative or indefinite integral. ∫(2x^5 - 5x + 8)dx

Explanation:

Step1: Apply sum - difference rule

The integral of a sum/difference is the sum/difference of integrals. So $\int(2x^{5}-5x + 8)dx=\int2x^{5}dx-\int5xdx+\int8dx$.

Step2: Apply constant - multiple rule

For $\int cf(x)dx = c\int f(x)dx$. We get $2\int x^{5}dx-5\int xdx + 8\int dx$.

Step3: Use power - rule for integration

The power - rule is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $\int x^{5}dx=\frac{x^{6}}{6}+C_1$, $\int xdx=\frac{x^{2}}{2}+C_2$, and $\int dx=x + C_3$. So $2\times\frac{x^{6}}{6}-5\times\frac{x^{2}}{2}+8x + C$ (where $C = C_1 + C_2 + C_3$).

Step4: Simplify the expression

$\frac{1}{3}x^{6}-\frac{5}{2}x^{2}+8x + C$.

Answer:

$\frac{1}{3}x^{6}-\frac{5}{2}x^{2}+8x + C$