`text(Differentials :)` Up to this point in our work, for `y = f (x)` we have regarded `dy/ dx` as a composite symbol for the derivative `f '(x)` , whose component parts, `dy` and `dx` , had no meaning by themselves. It is now convenient to modify this point of view and attach meaning to `dy` and `dx`, so that thereafter we can treat `dy/ dx` as though it were a fraction in fact as well as in appearance. We shall not however enter into any discussions on it. We shall only state that,
for a function of a single variable `y = f(x)` , the diffrential of `y` denoted by `dy` is the product of the derivative of `y` (with respect to `x`) and the diffrential of `x` denoted by `dx`. Thus,
`text(Differential of)` `y = f (x)` is `dy = f '(x)dx`.
For `y = x^4 , dy =4x^3dx`, or simply `d(x^4) = 4x^3 dx`. Thus
`d (sinx) = cosx dx, d (y^2) = 2y dy , d (tan u) = sec^2u du`.
`text(Integration As Anti-Derivative : )`
Simplest way to define integration is as an antiderivative the inverse of a derivative. Derivative of `sin x` is `cos x` then we may say that integral of `cos x` is `sin x`.
In general , if we consider
`quadquadquadquadquadquadquadd/dx f(x) =phi (x)`
or, using differentials `d f(x) = phi (x) dx`;
then an integral of `phi(x)` with respect to `x` or an integral of `phi (x) dx` is `f(x)` and symbolically, we write,
`quadquadquadquadquadint phi (x) dx = f(x)`
where the symbol `int` which is an elongated `S` (the first letter of the word sum, or, of the Latin word Swnma) is known as the sign of integration. Now we come to some formal definitions:
The actual process of finding the function, when its derivative or its differential is known, is called Integration as anti-derivative; the function to which the integration is applied is called Integrand and the function obtained as a result of integration is said to be lntegral. In the above case, `phi (x)` is the integrand and `f(x)` is the integral.
The process of integrating many ordinary functions is simple, but in general, integration is more involved than differentiation, as will be evident from future discussions.
`text(Summary: )`
If `d/dx [F(x) +C]= f(x) ` then `F(x) +C` is called an antiderivative of `f(x)` on `[a, b]` and is written as
`quadquadquadquadquadquadquadquadquadquadint f(x) dx =F(x) +C`
In this case we say that the function `f(x)` is integrable on `[a, b]`. Note that every function is not integrable.
e.g. `f(x) = [tt( (0,text(if) x in Q) ,(1, text(if) x notin Q) )` is not integrable in `[0, 1]`. Every function which is continuous on a closed and bounded interval is integrable.
However for integrability function `f(x)` may only be piece wise continuous in `(a, b)`
`text(Differentials :)` Up to this point in our work, for `y = f (x)` we have regarded `dy/ dx` as a composite symbol for the derivative `f '(x)` , whose component parts, `dy` and `dx` , had no meaning by themselves. It is now convenient to modify this point of view and attach meaning to `dy` and `dx`, so that thereafter we can treat `dy/ dx` as though it were a fraction in fact as well as in appearance. We shall not however enter into any discussions on it. We shall only state that,
for a function of a single variable `y = f(x)` , the diffrential of `y` denoted by `dy` is the product of the derivative of `y` (with respect to `x`) and the diffrential of `x` denoted by `dx`. Thus,
`text(Differential of)` `y = f (x)` is `dy = f '(x)dx`.
For `y = x^4 , dy =4x^3dx`, or simply `d(x^4) = 4x^3 dx`. Thus
`d (sinx) = cosx dx, d (y^2) = 2y dy , d (tan u) = sec^2u du`.
`text(Integration As Anti-Derivative : )`
Simplest way to define integration is as an antiderivative the inverse of a derivative. Derivative of `sin x` is `cos x` then we may say that integral of `cos x` is `sin x`.
In general , if we consider
`quadquadquadquadquadquadquadd/dx f(x) =phi (x)`
or, using differentials `d f(x) = phi (x) dx`;
then an integral of `phi(x)` with respect to `x` or an integral of `phi (x) dx` is `f(x)` and symbolically, we write,
`quadquadquadquadquadint phi (x) dx = f(x)`
where the symbol `int` which is an elongated `S` (the first letter of the word sum, or, of the Latin word Swnma) is known as the sign of integration. Now we come to some formal definitions:
The actual process of finding the function, when its derivative or its differential is known, is called Integration as anti-derivative; the function to which the integration is applied is called Integrand and the function obtained as a result of integration is said to be lntegral. In the above case, `phi (x)` is the integrand and `f(x)` is the integral.
The process of integrating many ordinary functions is simple, but in general, integration is more involved than differentiation, as will be evident from future discussions.
`text(Summary: )`
If `d/dx [F(x) +C]= f(x) ` then `F(x) +C` is called an antiderivative of `f(x)` on `[a, b]` and is written as
`quadquadquadquadquadquadquadquadquadquadint f(x) dx =F(x) +C`
In this case we say that the function `f(x)` is integrable on `[a, b]`. Note that every function is not integrable.
e.g. `f(x) = [tt( (0,text(if) x in Q) ,(1, text(if) x notin Q) )` is not integrable in `[0, 1]`. Every function which is continuous on a closed and bounded interval is integrable.
However for integrability function `f(x)` may only be piece wise continuous in `(a, b)`