Properties of definite integration:

1. Change of variable of integration is immaterial so long as limits of integration remain same that is

2. 

3. .

In general we break the limit 1^st at the points where f(x) is discontinuous and 2^nd at the points where definition of f(x) changes.

Illustration:  Evaluate , where [.] is greatest integer function.

Solution:             Let

                              Value of tan x at x = 5Π/12 is 2 + √3

                              Value of tan x at x = 0 is 0

                              The integers between 0 and 2 + √3 are 1, 2, 3

                              ∴ tan x = 1, tan x = 2, tan x = 3

                                => x = tan^-1 1, x = tan^-1 2, x = tan^-1 3 

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