Consider is `int_(0)^(pi//2) ln (sinx)dx` equal to ?
What is `int_(0)^(pi//2) ln (cos x) dx` equal to ?

Options

A. `I//2`
B. `I`
C. `2I`
D. `4I`

Answer

A

Explanation

Consider `I = int_(0)^(pi) ln(sinx) dx`
`I = int_(0)^(pi) ln(sinx) dx`
`=2int_(0)^(pi/2)ln (sinx)dx"……….."(i)`
`[ :' int_(0)^(2a) f(x) dx , " if " f(2a-x) = f(x)]`
`= 2int_(0)^(pi/2) ln [sin(pi/2 - x)]dx`
`( :' int_(0)^(a) f(x) dx= int_(0)^(a) f(a-x)dx)`
`= 2int_(0)^(pi/2) ln(cosx) dx"......"(ii)`
From eq. (ii), we have
`I = 2int_(0)^(pi/2) ln (cosx) dx`
`rArr int_(0)^(pi/2) ln (cos x) dx = 1/2 I`