If `y=y(x)` and `(2+sinx)/(y+1)((dy)/(dx))=-cosx ,y(0)=1,` then `y(pi/2)` equals (a) `( b ) (c) (d)1/( e )3( f ) (g) (h)` (i) (b) `( j ) (k) (l)2/( m )3( n ) (o) (p)` (q) (c) `( r ) (s)-( t )1/( u )3( v ) (w) (x)` (y) (d) 1

Options

A. `1/3`
B. `2/3`
C. `-1/3`
D. `1`

Answer

A

Explanation

Given `(dy)/(y+1)=(-cosx)/(2+sinx)dx`
`impliesint(dy)/(y+1)=-int(cosx)/(2+sinx)dx`
`implieslog(y+1)=-log(2+sinx)+logc`
when `x=0,y=1`
`implies c=4`
`:.y+1=4/(2+sinx)`
At `x=(pi)/2,y+1=4/(2+1)`
`impliesy=1/3`