Karnataka Board PUCPUC Science 2nd PUC Class 12

Some Properties of Indefinite Integral




The explain the some properties of indefinite integrals. 
I)  The process of differentiation and integration are inverses of each other in the sense of the following results :
`d/(dx) int` f(x)dx = f(x)
and `int f '(x) dx = f(x) + C` , where C is any arbitrary constant.

Proof: Let F be any anti derivative of f, i.e.,
`d/(dx)` F(x) = f(x)
Then `int` f(x)dx = F(x) +C
Therefore `d/(dx) int` f(x) dx = `d/(dx)`(f(x)+C)
= `d/(dx)` F(x)=f(x)
Similarly, we note that
f'(x) = `d/(dx)` f(x)
and hence `int`f'(x) dx = f(x) +C
where C is arbitrary constant called constant of integration. 

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

Proof: Let f and g be two functions such that
`d/(dx) int f(x)dx = d/(dx) int g(x)dx`

or `d/dx [int f(x) dx - int g(x) dx]` = 0 

Hence ∫f(x) dx - ∫g(x) dx = C, where C is any real number

or `int f(x)dx = int g(x)dx + C `

So the families of curves `{int f(x) dx + C_1, C_1 ∈ R}`
and `{int g(x) dx + C_2 , C_2 ∈ R}` are identical.
Hence, in this sense, `int f(x) dx ` and `int g(x) dx` are equivalent.

III) ∫[f(x) + g(x)]dx = ∫f(x)dx +  ∫ g(x) dx

Proof:  By Property (I), we have
`d/(dx)int[f(x) + g(x)dx] = f(x) +g(x)`     ...(1)
 On the otherhand, we find that
`d/(dx)[ int  f(x) dx + int g(x) dx] = d/(dx) int f(x) dx + d/(dx) int g(x) dx`
=f(x) + g(x)                                                  ...(2)

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

`int (f(x) + g(x))dx = int f(x) dx + int  g(x) dx .` 

IV)  For any real number k, `int  k f(x) dx = k int f(x) dx`

Proof:  By the Property (I),
`d/(dx) int k f(x) dx = k f(x).`
Also `d/(dx) [k int f(x)dx] = k d/(dx) int f(x) dx` = k f(x)
 Therefore, using the Property (II), we have `int k f(x) dx = k int f(x) dx .` 

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

`int [k_1f_1(x) + k_2f_2(x) + ...+k_nf_n (x)] dx`   
= `k_1 int f_1(x) dx +k_2 int f_2 (x) dx + ... + k_n int f_n (x) dx`

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