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

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

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`

Text Solution

Verified by Experts

The correct Answer is:
B
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`)
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