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Show that int(0)^(pi//2) log (sin x) dx ...

Show that `int_(0)^(pi//2) log (sin x) dx = - pi/2 log 2`

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int_(0)^(pi//2) log (tan x ) dx=

int_(0)^(pi//2)log (sec x) dx=

int_(0)^(pi//2) log (cotx ) dx=

int_(0)^((pi)/(2))log(sin2x)dx

Show that int_(0)^((pi)/(2))log(sin2x)dx=-(pi)/(2)(log2)

int_(0)^( pi)cos2x*log(sin x)dx

int_(0)^((pi)/(2))log(cos x)dx=

int_(0)^((pi)/(2))log(tan x)*dx