Question

Choose the correct answerThe value of `int_0^(pi/2) log((4+3sinx)/(4+3cosx))dx`(A) 2 (B) `3/4` (C) 0 (D) -2

Answer 1

Here, we will use the property,
`int_0^a f(x) dx = int_0^a f(a-x) dx`
Let `I = int_0^(pi/2) log((4+3sinx)/(4+3cosx)) dx->(1)`
Using the above property,
`I = int_0^(pi/2) log((4+3sin(pi/2-x))/(4+3cos(pi/2-x))) dx`
`=>I = int_0^(pi/2) log((4+3cosx)/(4+3sinx) )dx ->(2)`
Now, adding (1) and (2),
`2I = int_0^(pi/2) log((4+3sinx)/(4+3cosx)) dx + int_0^(pi/2) log((4+3cosx)/(4+3sinx) )dx`
`=>2I = int_0^(pi/2) [log((4+3sinx)/(4+3cosx))+log((4+3cosx)/(4+3sinx) )]dx`
`=>2I = int_0^(pi/2) [log((4+3sinx)/(4+3cosx)**(4+3cosx)/(4+3sinx) )) dx`
`=>2I = int_0^(pi/2) log1 dx`
`=>2I = int_0^(pi/2) 0 dx`
`=>2I = 0`
`=>I = 0`