Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12

Some Properties of Indefinite Integral

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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 intf'(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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Integrals part 5 (Properties of indefinite integrals) [00:09:34]
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