Class 12 Maths Integrals Integration as an Inverse Process of Differentiation

Integration as an Inverse Process of Differentiation

Integration is the inverse process of differentiation. Let f(x) be a function. Then the collection of all primitives is called the indefinite integral of f(x) and denoted by ʃ f(x) dx.

Thus, d[ф(x) + C]/dx = f(x) => ʃ f(x) dx = ф(x) + C

Where ф(x) is primitive of f(x) and C is an arbitrary constant known as constant of integration.

The following is a list to find integrals of other functions.

Some properties of indefinite integral

(i) The process of differentiation and integration are inverses of each other in the sense of the following results :

d[ʃ f(x) dx]/dx = f(x)

and  ʃ f’(x) dx = f(x) + C, where C is any arbitrary constant.

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

i.e. d[ʃ f(x) dx]/dx = d[ʃ g(x) dx]/dx

=> d[ʃ f(x) dx - ʃ g(x) dx]/dx = 0

=> ʃ f(x) dx - ʃ g(x) dx = C, where C is any real number

=> ʃ f(x) dx = ʃ g(x) dx + C

So, the families of curve [ʃ f(x) dx + C1, C1 є R] and [ʃ g(x) dx + C2, C2 є R] are identical.

Hence, ʃ f(x) dx and ʃ g(x) dx are equivalent.

(iii) ʃ [f(x) + g(x)] dx = ʃ f(x) dx + ʃ g(x) dx

(iv) For any real number k, ʃ [k * f(x)] dx = k * ʃ f(x) dx

(v) ʃ [k1 * f1(x) + k2 * f2(x) + . . . . .+ kn * fn(x)] dx = k1 * ʃ f1(x) dx + k2 * ʃ f2(x) dx + . . . . . .+ kn * ʃ fn(x) dx

Problem: Find the integral of the following functions:

(a) ʃ (ax2 + bx + c) dx  (b) ʃ (2x2 + ex) dx

Solution:

(a) ʃ (ax2 + bx + c) dx = ʃ ax2 dx + ʃ bx dx + ʃ c dx

= aʃ x2 dx + b ʃ x dx + c ʃ dx

= ax3/3 + bx2/2 + cx + C

So, ʃ (ax2 + bx + c) dx = ax3/3 + bx2/2 + cx + C

(b) ʃ (2x2 + ex) dx = ʃ 2x2 dx + ʃ ex dx

= 2ʃ x2 dx + ʃ ex dx

= 2x3/3 + ex + C

So, ʃ (2x2 + ex) dx = 2x3/3 + ex + C

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