find the most general antiderivative of the function. 11) f(x)=1/2x^2 - 2x + 6

Explanation:

Step1: Recall power - rule for antiderivatives

The antiderivative of $x^n$ is $\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For the function $f(x)=\frac{1}{2}x^{2}-2x + 6$, we find the antiderivative term - by - term.

Step2: Antiderivative of $\frac{1}{2}x^{2}$

Using the power - rule, $\int\frac{1}{2}x^{2}dx=\frac{1}{2}\times\frac{x^{2 + 1}}{2+1}=\frac{1}{6}x^{3}$.

Step3: Antiderivative of $-2x$

Using the power - rule, $\int(-2x)dx=-2\times\frac{x^{1 + 1}}{1+1}=-x^{2}$.

Step4: Antiderivative of 6

Since 6 can be written as $6x^{0}$, $\int6dx=6\times\frac{x^{0 + 1}}{0+1}=6x$.

Step5: Combine and add the constant of integration

The most general antiderivative $F(x)$ of $f(x)$ is $F(x)=\frac{1}{6}x^{3}-x^{2}+6x + C$, where $C$ is an arbitrary constant.

Answer:

$F(x)=\frac{1}{6}x^{3}-x^{2}+6x + C$