Q. 6 - Calculus

Question

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.

• the indefinite integral of a continuous function f

Step-by-Step Solution

Answer

Ans: $F\left(x\right)=\int f\left(x\right)dx+A$ is called indefinite integral with respect to x

Step 1. Given information:

The indefinite integral of a continuous function f

Step 2. Defining with an algebraic example :

$\int f\left(x\right)dx=F\left(x\right)+C$ , where the function $F\left(x\right)$ is antiderivative of the function $f\left(x\right)$.

For example, the antiderivative of the function $f\left(x\right)=3x^2$ is $F\left(x\right)=x^3$,

It follows that all antiderivative of $f\left(x\right)=3x^2$ are of the form $x^3+C$, where C is a constant and therefore, $\int 3x^{2}dx=x^3+C$.

The indefinite integral of function $f\left(x\right)=3x^2$ is defined as $F\left(x\right)=\int f\left(x\right)dx+A$ or $F\left(x\right)=x^3+A$. The constant A can be computed with the initial condition $F\left(c\right)$.

The constant has different values for different x i.e., it provides indefiniteness to the integral which is why the integral $F\left(x\right)=\int f\left(x\right)dx+A$ is called indefinite integral with respect to x.