Class 12 Maths Integrals Evaluation of Definite Integrals by Substitution

Evaluation of Definite Integrals by Substitution

To evaluate aʃb f(x) dx by substitution, the steps could be as follows:

  1. Consider the integral without limits and substitute, y = f (x) or x = g(y) to reduce the given integral to a known form.
  1. Integrate the new integrand with respect to the new variable without mentioning the constant of integration.
  1. Resubstitute for the new variable and write the answer in terms of the original variable.
  2. Find the values of answers obtained in (3) at the given limits of integral and find the difference of the

values at the upper and lower limits.

Problem: Evaluate the following integrals using substitution method

(a) ʃ01 [x/(x2 + 1)] dx      (b) ʃ0π/2 [√(sin ф) * cos5 ф] dф

Solution:

(a) Let x2 + 1 = t

=> 2x dx = dt

=> x dx = dt/2

When x = 0, t = 1 and when x = 1, t = 2

So, ʃ01 [x/(x2 + 1)] dx = (1/2)ʃ12 dt/t

                                     = (1/2)[log t]12

                                     = (1/2)[log 2 – log 1]

                                     = (log 2)/2

 

(b) Let I = ʃ0π/2 [√(sin ф) * cos5 ф] dф

               = ʃ0π/2 [√(sin ф) * cos4 ф * cos ф] dф

Let sin ф = t

=> cos ф dф = dt

When ф = 0, t = 0 and when ф = π/2, t = 1

So, I = ʃ01 [√t * (1 – t2)2] dt

        = ʃ01 [√t * (1 + t4 - 2t2)] dt

        = ʃ01 [t1/2 + t9/2 - 2t5/2] dt

        = [t3/2/(3/2) + t11/2/(11/2) - 2t7/2/(7/2)]01

        = 1/(3/2) + 1/(11/2) - 2/(7/2)

        = 2/3 + 2/11 – 4/7

        = (154 + 42 - 132)/231

        = 64/231

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