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 $\smallint [{x^n}] = \dfrac{{{x^{(n + 1)}}}}{{(n + 1)}}$
$\smallint [f'(x)] = {x^2} + c$ 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 :
$\smallint {\text{ }}1{\text{ }}dx{\text{ }} = {\text{ }}x{\text{ }} + {\text{ }}C.$
$\smallint {\text{ }}a{\text{ }}dx{\text{ }} = {\text{ }}ax + {\text{ }}C.$
$\smallint {x^n}dx = [\dfrac{{{x^{(n + 1)}}}}{{(n + 1)}}] + C;n! = 1$
$\smallint {\text{ }}sin{\text{ }}x{\text{ }}dx{\text{ }} = {\text{ }}-{\text{ }}cos{\text{ }}x{\text{ }} + {\text{ }}C.$
$\smallint {\text{ }}cos{\text{ }}x{\text{ }}dx{\text{ }} = {\text{ }}sin{\text{ }}x{\text{ }} + {\text{ }}C.$
$\smallint se{c^2}xdx = \tan x + C.$
$\smallint cose{c^2}xdx = - \cot x + C$