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Question
Evaluate the following integrals using properties of integration:
`int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan x)) "d"x`
Solution
Let I = `int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan x)) "d"x` .......(1)
`int_"a"^"b" f(x) "d"x = int_"a"^"b" f("a" + "b" - x) "d"x`
I = `int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan (pi/8 + (3pi)/8 - x))) "d"x`
= `int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan(pi/2 - x))) "d"x`
= `int_(pi/8)^((3pi)/8) 1/(1 + sqrt(cot x)) "d"x`
= `int_(pi/8)^((3pi)/8) 1/(1 + 1/sqrt(tan x)) "d"x`
I = `int_(pi/8)^((3pi)/8) sqrt(tanx)/(1 + sqrt(tan x)) "d"x` ........(2)
Add (1) + (2)
2I = `int_(pi/8)^((3pi)/8) (1 + sqrt(tan x))/(1 + sqrt(tan x)) "d"x`
= `int_(pi/8)^((3pi)/8) "d"x`
= `[x]_(pi/8)^((3pi)/8)`
= `(3pi)/8- pi/8`
2I = `(2pi)/8 = (pi/4)`
I = `pi/8`
`int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan x)) "d"x = pi/8`
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