Question

What do you mean by integral value ?

Answer 1

Hint: We have to solve this question by stating the definition of integral value . We also state the two types of integrals . We will also give examples of the integral values . We will also give the places where the integral values can be used . We will also give some of the formulas to integrate a function . We can also give the relation between differentiation and integral values .

Complete step-by-step answer:
Definition of integral values :
In general term integral value means the value obtained after integrating or adding the terms of a function which is divided into an infinite number of terms .

Types of integral values :
(1) Indefinite integral
(2) Definite integral

(1) indefinite integral : The anti derivative or integration of a function which is not integrated for any particular value or limit . It is just defined in general terms . The integration of the terms consist of an integral constant which can have any value . The integral constant is added as the differentiation of a constant term is zero . So , while integrating we can’t calculate the value of that constant in an indefinite integral so we have to add an integral constant as the integration is not defined for a limit .
Example : let the original function : $f(x)=x^2+100$
Now , if we differentiate the function $$f\left(x\right)$$ with respect to x we get 2x,
Now finding the antiderivative of, we get
Integrating the function using the formula $\int [x^n]={\displaystyle \frac{x^{(n+1)}}{(n+1)}}$
$\int [f^{\prime }(x)]=x^2+c$ Where ‘c’ is the integral constant . Which we can’t determine .

(2) definite integral
The anti derivative or integration of a function which is integrated for any particular value or limit . It is just defined in general terms . The integration of the terms does not consist of an integral constant . When a function is integrated for a particular limit we get a definite value of the function .
The concept of integral values is used to find the values of area or volume of a shape or a region formed by a set of equations . We can also determine the displacement or acceleration of a function of motion .
The area or volume can be calculated by using the various integral formulas and integrating the function for a limit. Putting the limit gives us the exact value ( integral value ) of area or volume .

Note: Various integration formulas are :
$$\int 1dx=x+C.$$
$$\int adx=ax+C.$$
$\int x^{n}dx=[{\displaystyle \frac{x^{(n+1)}}{(n+1)}}]+C;n!=1$
$$\int sinxdx=-cosx+C.$$
$$\int cosxdx=sinx+C.$$
$\int se{c}^{2}xdx=\tan x+C.$
$\int cose{c}^{2}xdx=-\cot x+C$