An integral which is not having any upper and lower limit.
F(x) is the way function f(x) is integrated and it is represented by:
Where in respect to x the integral of f(x) is on the R.H.S.
primitive or anti-derivative is termed as F(x)
Integrand is termed as f(x)
Integrating agent is termed as dx
The constant of integration is an arbitrary constant termed as C
The variable of integration is termed as x
Properties of Indefinite Integrals
Let us now look into some properties of indefinite integrals.
- Differentiation and integration are inverse processes of each other since:and
Where C is any arbitrary constant. Let us now prove this statement.
Proof: Consider a function f such that its anti-derivative is given by F, i.e.
On differentiating both the sides with respect to x we have,
Since derivative of any constant function is zero, therefore
The derivative of a function f in x is given as f’(x), therefore
Where, C is an arbitrary constant called as the constant of integration.
- Two indefinite integrals having the same derivative have the same family of integrals or curves and therefore they are equivalent.
Consider two functions f and g in x such that:
On integrating both sides, we get
where C is any real number.
- The integral of sum of two functions is equal to the sum of integrals of the given functions, i.e.,
Proof: By the first property of integrals we have,
Therefore from equation (1) and (2) we have,
- For any real value of p,
Proof: From property 1 we can say that
From property 2 we can say that
- For a finite number of functions f1, f2…. fn and the real numbers p1, p2…pn,
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