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Evaluate int(0)^((pi)/2)log((4+3sinx)/(4...

Evaluate `int_(0)^((pi)/2)log((4+3sinx)/(4+3cosx))dx`

Text Solution

Verified by Experts

The correct Answer is:
C
`"Let I "= int_(0)^(pi//2) log ((4+3 sin x)/(4+3 cos x)) dx" "……(1)`
` rArr 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]`
`rArr I= int_(0)^(pi//2) log ((4+3 cos x)/(4+3 sin x))dx" "....(2)`
Adding equations (1) and (2)
`2I = int_(0)^(pi//2) [log ((4+3 sin x)/(4+3 sin x)) `
`+log ((4+3 cos x)/(4+3 sin x))]dx`
`rArr 2I =int_(0)^(pi//2) log ((4+3 sin x)/(4+3 cos x)xx(4+3 cos x)/(4+3 sin x)dx)`
`rArr 2I =int_(0)^(pi//2) log 1 dx`
`rArr 2I =int_(0)^(pi//2) 0dx`
`rArr I=0`
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