`int \ sin x \ d(cos x)` is equal to

Options

A. `1/2 sin 2x +x/2 +c`
B. `1/2 sin 2x -x/2 +c`
C. `2 sin 2x +c`
D. `sin x +cos x`

Answer

B

Explanation

Let `I=int sinx d(cos x)`
Put `cos x=t`
`impliesd(cos x)=dt`
`:. I = int sqrt(1-t^(2))dt`
`=t/2sqrt(1-t^(2))+1/2sin^(-1)(t)+c_(1)`
`=(cos x)/2sin x+1/2 sin^(-1)(cosx)+c_(1)`
`=1/4sin 2x+1/2 sin^(-1)[sin((pi)/2-x)]+c_(1)`
`=1/4 sin 2x-x/2 +c`
( let `(pi)/4+c_(1)=c`)