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Question
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is ______.
Options
2
`3/4`
0
- 2
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Solution
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is 0.
Explanation:
Let I `= int_0^(pi//2) log ((4 + 3 sin x)/(4 + 3 cos x)) "dx"`
Also, `I = int_0^(pi/2) log [(4+3 sin (pi/2 - x))/(4 + 3 cos (pi/2 - x))] dx`
`[∵ int_0^a f (x) dx = int_0^a f (a - x) dx]`
⇒ ` I = int_0^(pi/2) log [(4+3 cos x)/(4+3 sin x)] dx`
⇒ `I = - int_0^(pi/2) log [(4+3sinx)/(4+3cosx)] dx`
⇒ I = -I
⇒ 2I = 0
⇒ I = 0
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