please answer each of the following three questions in the order as it is and reply to at least two…

Brief Explanations:

1. An antiderivative of a function $f(x)$ is a function $F(x)$ such that $F'(x)=f(x)$. An integral can be a definite integral $\int_{a}^{b}f(x)dx$ which gives a number (the net - signed area under the curve $y = f(x)$ from $x=a$ to $x = b$) or an indefinite integral $\int f(x)dx=F(x)+C$ where $F(x)$ is an antiderivative and $C$ is the constant of integration. The indefinite integral represents a family of antiderivatives.
2. Let the function be $f(x)=x^2$. Using the power rule for antiderivatives $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), the antiderivative of $f(x)=x^2$ is $F(x)=\frac{x^{3}}{3}+C$.

Answer:

1. An antiderivative is a function whose derivative is the given function. An indefinite integral is a family of antiderivatives (with a constant of integration), and a definite integral is a number representing an area.
2. For the function $f(x)=x^2$, its antiderivative is $F(x)=\frac{x^{3}}{3}+C$.