`((int (x)/(2))/(-(x)/(2)))log((2-sinx)/(2+sin x))` dx is equal to

Options

A. 1
B. 3
C. 2
D. 0

Answer

D

Explanation

We have, `i=(underset(-x)overset((pi)/(2))int)/(2) log ((2-sin x)/(2+sin x))dx`
`"Let " f(x)=log ((2-sinx)/(2+sin x))`
Then, f(-x)=`log ((2-sin(-x))/(2+sin(-x)))`
`=log ((2+sin x)/(2-sinx))=log ((2-sinx)/(2+sin x))^(-1)`
`=-log ((2-sin x)/(2+sin x))=-f(x)`
Then, f(x) is an odd function.
`therefore (underset(pi)overset((pi)/(2))int (2) f(x)dx=0`
`[therefore ` If f(x) is an odd function, then `underset(-a)overset(a) f(x)dx=0`