`int_(0)^((pi)/(2))log((4+3sinx)/(4+3 cos x))dx` का मान है :

Explanation

माना `I=int_(0)^(pi//2)log((4+3sinx)/(4+3 cos x))dx" …(1)"`
`rArr" "I=int_(0)^(pi//2)log[(4+3sin(pi//2-x))/(4+3cos(pi//2-x))]dx`
`" "[because 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)"`
समीकरण (1 ) और (2 ) को जोड़ने पर
`2I=int_(0)^(pi//2)[log((4+3sinx)/(4+3 sinx))+log((4+3 cosx)/(4+3 sinx))]dx`
`rArr" "2I=int_(0)^(pi//2)log((4+3sinx)/(4+3cosx)xx(4+3 cosx)/(4+3 sinx)dx)`
`rArr" "2I=int_(0)^(pi//2)log1 dx`
`rArr" "2I=int_(0)^(pi//2)0dx" "(because log 1=0)`
`rArr" "I=0`