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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`