Q. 7 - 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.

• what we mean when we refer to an integrand 

Step-by-Step Solution

Answer

Ans: $\frac{dF\left(x\right)}{dx}=3x^2$  is integrand of the integral  $I=\int 3x^{2}dx+A$

Step 1. Given information:

Integrand of an integral .

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 coefficient of $\mathrm{dx}$ or the function $f\left(x\right)$ of which the integral is sought or the derivative of the function $F\left(x\right)$ is called the integrand of the integral$\int f\left(x\right)dx=F\left(x\right)+C$

The indefinite integral of function $f\left(x\right)=3x^2$ is definedas $F\left(x\right)=\int f\left(x\right)dx+A$, or $F\left(x\right)=x^3+A$. Here, $3x^2$ or $\frac{dF\left(x\right)}{dx}=3x^2$ is integrand of the integral $I=\int 3x^{2}dx+A$.