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int(sin(101x).sin^(99)x)dx equals...

`int(sin(101x).sin^(99)x)dx` equals

A

`(sin(100x)(sinx)^(100))/(100)+C`

B

`(cos(100x)(sinx)^(100))/(100)+C`

C

`(cos(100x)(cosx)^(100))/(100)+C`

D

`(cos(100x)(cosx)^(100))/(100)+C`

Text Solution

Verified by Experts

The correct Answer is:
A
`I=int(sin(100x)+x).(sinx)^(99)dx`
`=int[(sin(100x)cosx+cos100x.sinx)(sinx)^(99)]dx`
`=int underset("I")(ubrace(sin(00x)))underset("II")(ubrace(cosx.(sinx)^(99)))dx+intcos(100x).(sinx)^(100)dx`
`=(sin(100x)(sinx)^(100))/(100)-(100)/(100)intcos(100x)(sinx)^(100)dx+int cos(100x)(sinx)^(100)dx`
`=(sin(100x)(sinx)^(100))/(100)+C`
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