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प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/4) log (1+ tan x) dx`
Evaluate:
`int_0^(pi/4) log (1+ tan x) dx`
उत्तर
Let I = `int_0^(pi/4) log (1 + tan x) dx` ....(1)
∴ I = `int_0^(pi/4) log [1 + tan (pi/4 - x)] dx` `...[int_0^a f(x) dx = int_0^a f(a - x) dx]`
⇒ I = `int_0^(pi/4) log {1 + (tan pi/4 - tan x)/(1 + tan pi/4 tan x)}dx`
⇒ I = `int_0^(pi/4) log {1 + (1 - tan x)/(1 + tan x)} dx`
⇒ I = `int_0^(pi/4) log 2/((1 + tan x)) dx`
⇒ I = `int_0^(pi/4) log 2 dx - int_0^(pi/4) log (1 + tan x) dx`
⇒ I = `int_0^(pi/4) log 2 dx - I` ...[From (1)]
⇒ 2I = `[x log 2]_0^(pi/4)`
⇒ 2I = `pi/4 log 2`
⇒ I = `pi/8 log 2`
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