Class 12 Maths Integrals Methods of Integration

Methods of Integration

There are certain methods for solving the integration:

  1. Integration by Substitution
  2. Integration using Partial Fractions
  3. Integration by Parts
  4. Integration by Substitution

The given integral ʃ f(x) dx can be transformed into another form by changing the independent variable x to t by substituting x = g (t).

Let I = ʃ f(x) dx

Put x = g(t) so that dx/dt = g′(t).

We write dx = g′(t) dt

Thus I = ʃ f(x) dx = ʃ f{g(t)} * g’(t) dt

Problem: Integrate the following functions:

(a) 2x/(1 + x2)   (b) (log x)2/x

Solution:

(a) Let 1 + x2 = t

=> 2x dx = dt

Now, ʃ 2x/(1 + x2) dx = ʃ dt/t

                                      = log|t| + C

                                      = log|1 + x2| + C

                                      = log(1 + x2) + C

(b) Let log x = t

=> dx/x = dt

Now, ʃ [(log x)2/x] dx = ʃ t2 dt     

                                      = t3/3 + C

                                      = (log|x|)3/3 + C

There are some important trigonometries integral which are listed below:

(i) ʃ tan x dx = log|sec x| + C  

(ii) ʃ cot x dx = log|sin x| + C

(iii) ʃ sec x dx = log|sec x + tan x| + C

(iv) ʃ cosec x dx = log|cosec x – cot x| + C

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