`\color{green} ✍️` Process of finding anti derivatives is called integration .
`\color{green} ✍️` There are two forms of the integrals, e.g., indefinite and definite integrals, which together constitute the Integral Calculus.

`\color{green} ✍️` Integration is the inverse process of differentiation.

`\color{green} ✍️` Instead of differentiating a function, we are given the derivative of a function and asked to find its primitive, i.e., the original function. Such a process is called integration or anti differentiation.

`\color{green} ✍️` If there is a function F such that `d/(dx) F(x) = f(x) , AA x ∈ I` (interval), then for any arbitrary real number C, (also called constant of integration)

`color{orange}{d/(dx) [F(x) +C ] = f(x) , x ∈ I"`


`\color{green} ✍️` A new symbol, namely, ∫ f (x) dx which will represent the entire class of"`
`\color{green}" anti derivatives read as the indefinite integral of f with respect to x

Symbolically, we write `color{orange} (∫ f (x) dx = F (x) + C.)`

Notation Given that ` (dy)/(dx)= f(x)` , we write `y = ∫ f (x) dx .`

- we mention below the following symbols/terms/phrases with their meanings as given in the Table.

● Let f (x) = 2x, Then `color{orange}(∫ f (x) dx = x^2 + C)` . For different values of C, we get different integrals. But these integrals are very similar geometrically.

● `y = x^2 + C,` where C is arbitrary constant, represents a family of integrals. By assigning different values to `C,` we get different members of the family.

● Clearly, for `C = 0,` we obtain `y = x^2 ,` a parabola with its vertex on the origin. The curve `y = x^2 + 1` for `C = 1` is obtained by shifting the parabola `y = x^2` one unit along y-axis in positive direction. For `C = – 1, y = x^2 – 1` is obtained by shifting the parabola `y = x^2
` one unit along y-axis in the negative direction and so on.

● So `color{orange}{∫ f (x) dx = x^2 + C}" represents a family of curves."` The different values of C will correspond to different members of this family and these members can be obtained by shifting any one of the curves parallel to itself.

● `"This is the geometrical interpretation of indefinite integral."`

`"Property 1 :"`

`color{blue}"The process of differentiation and integration are inverses of each other"`

`color{orange} {=> d/(dx) ∫ f(x) dx = f(x)}` and ` color{orange} {∫ f' (x) dx = f(x) +C}` ,

where `C ` is any arbitrary constant.

`color{red}{ul{" Proof :"` Let F be any anti derivative of f, i.e.,

`color{orange} {d/(dx) F(x) = f(x)}`

Then ` color{orange} {∫ f (x) dx = F(x) +C}`

`therefore` `color{orange} {d/(dx) ∫ f(x) dx = d/(dx) (F(x) +C )}`

` color{orange} { = d/(dx) F(x) = f(x)}`

Similarly, we note that

`color{orange} {f'(x) = d/(dx) f(x)}` and hence` color{orange} {∫ f ′(x) dx = f (x) + C}`

where C is arbitrary constant called constant of integration.

`color{blue}"Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent. "`


`color{red}{ul{" Proof :"` Let f and g be two functions such that

`color{orange} {d/(dx) ∫ f(x) dx = d/(dx) ∫ g (x) dx}`

or `color{orange} {d/(dx) [ ∫ f(x) dx- ∫ g(x) dx ] = 0}`

Hence ` color{orange} {∫ f(x) dx - ∫ g (x) dx = C}` , where C is any real number

or ` color{orange} {∫ f (x) dx = ∫ g (x) dx + C}`

So the families of curves ` color{green} { ∫ f(x) dx +C_1 ,C_1 ∈ R }`

and ` color{green} { ∫ g(x) dx +C_2, C_2 ∈ R }`are identical

Hence, in this sense , `color{orange} {∫ f (x) dx}` and ` color{orange} {∫ g (x) dx} ` are equivalent.

`color{blue}( ∫ [ f(x) +g(x) ] dx = ∫ f(x) dx + ∫ g(x) dx)`

`color{red}{ul{" Proof :"` By Property (I), we have

`color{orange} {d/(dx) [ ∫ [ f(x) + g(x)] dx ] = f(x) + g(x)}` ........(1)

On the otherhand, we find that

`color{orange} {d/(dx) [ ∫ f(x) dx + ∫ g(x) dx ] = d/(dx) ∫ f(x) dx + d/(dx) ∫ g(x) dx}`

`color{orange} {= f(x) + g(x)}` ........(2)

Thus, in view of Property (II), it follows by (1) and (2) that

`color{orange} { ∫ ( f(x) +g(x) ) dx = ∫ f(x) dx + ∫ g(x) dx}`

For any real number k, `color{blue}( ∫ k f(x) dx= k ∫ f(x) dx)`

Proof By the Property (I), `color{orange} {d/(dx) ∫ k f(x) dx = k f(x)}`

Also `color{orange} {d/(dx) [ k ∫ f(x) dx ] = k d/(dx) ∫ f(x) dx = k f(x)}`

Therefore, using the Property (II), we have ` color{orange} {∫ f(x) dx = k ∫ f(x) dx}` .

- By Properties (III) and (IV) can be generalised to a finite number of functions `color{green} {f_1, f_2, ..., f_n}` and the real numbers, `color{green} {k_1, k_2, ..., k_n}` giving

`color{orange} { ∫ [k_1 f_1 (x) + k_2 f_2 (x) + .....+ k_n f_n (x) ] dx}`

`color{orange} { = k_1 ∫ f_1 (x) dx + k_2 ∫ f_2(x) dx + .....+ k_n ∫ f_n (x) dx}` .

(i) We see that if F is an anti derivative of f, then so is `F + C,` where `C` is an constant. Thus, if we know one anti derivative `F` of a function `f,` we can write down an infinite number of anti derivatives of `f` by adding any constant to F expressed by `F(x) + C, C ∈ R.` In applications, it is often necessary to satisfy an additional condition which then determines a specific value of `C` giving unique anti derivative of the given function.

(ii) Sometimes, F is not expressible in terms of elementary functions viz., polynomial, logarithmic, exponential, trigonometric functions and their inverses etc. We are therefore blocked for finding `∫f (x) dx.` For example, it is not possible to find `∫e^((-x)^2) dx` by inspection since we can not find a function whose derivative is `e^((-x)^2).`

(iii) When the variable of integration is denoted by a variable other than `x,` the integral formulae are modified accordingly. For instance

`int y^4 dy = y^5/5 + c`