Question

Some Properties of Indefinite Integral

Answer 1

Property 1. 

Differentiation and integration are inverse process to each other.

i.e \(\frac{d}{dx}\) ∫f(x)dx = f'(x)

and ∫f'(x) dx = f(x) + C, where C is any constant which is called constant of integration.

Proof:

Let anti-derivative of f is F

i.e., \(\frac{d}{dx}\) F(x)dx = f(x)

Then ∫f(x)dx = F(x) + C

∴ \(\frac{d}{dx}\) ∫f(x) dx = \(\frac{d}{dx}\) (F(x) + C) = \(\frac{d}{dx}\) (F(x)) = f(x)

We see that f'(x) = \(\frac{d}{dx}\) f(x)

∴∫f'(x)dx = f(x) + C,

where C is any constant which is called constant of integration.

Property 2:

The integral of the product of a constant and a function is equal to the product of the constant and the integral of the function.

i.e., ∫kf(x) dx = k ∫f(x) dx, where k is a constant.

Proof:

\(\frac{d}{dx}\) ∫kf(x) dx = k \(\frac{d}{dx}\) ∫f(x) dx = kf(x)

and \(\frac{d}{dx}\) [k∫f(x) dx] = k \(\frac{d}{dx}\) ∫f(x) dx = kf(x)

∴ ∫k f(x) dx = k∫f(x) dx

Property 3:

(A) The integral of the sum of two function is equal to the sum of the integrals of the two functions.

i.e., ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx

(B) The integral of the difference of two function is equal to the difference of the integrals of the two functions.

i.e., ∫[f(x) - g(x)] dx = ∫f(x) dx - ∫g(x) dx

Proof: (A) {∫ [fix) + g(x)]dx} = f(x) + g(x) ...(1)

Again, \(\frac{d}{dx}\) [∫f(x) dx + ∫g(x) dx]

= \(\frac{d}{dx}\) ∫f(x) dx + \(\frac{d}{dx}\) ∫g(x) dx

= f(x) + g(x) ...(2)

From (1) and (2), derivative of functions are same.

Hence, ∫ [f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx

Similarly, (B) can be proved.

Generalization: The above results can be generalized to the form:

∫[k₁f₁(x) ± k₂f₂(x) ... ± k_nf_n{x)} dx = k₁∫f₁(x) dx ± k₂∫f₂(x) dx... ±f_n∫f_n(x) dx

Property 4 :

Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent.

Proof:

Let f and g are two functions such that

= \(\frac{d}{dx}\) ∫f(x)dx = \(\frac{d}{dx}\) ∫g(x)dx

or = \(\frac{d}{dx}\) [∫f(x) dx - ∫g(x) dx] = 0

∴ ∫f(x) dx - ∫g(x) dx = C

where C is a real number

or ∫f(x) dx = ∫g(x) dx + C

Hence, family of curves ∫f(x) dx + C₁ and ∫g(x) dx + C₂ are identical where C₁ and C₂ are real numbers.

∴ ∫f(x) dx and ∫g(x) dx are equivalent.