If I(n)=int cos^(n)x dx. Prove that I(n...

If `I_(n)=int cos^(n)x dx`. Prove that `I_(n)=(1)/(n)(cos^(n-1)x sinx)+((n-1)/(n))I_(n-2)`.

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`I_(n)=int cos^(n)x dx`
`=cos^(n-1)x intcosx dx+(n-1)int(sin^(2)x)cos^(n-2)x dx`
`=(cos^(n-1) x sinx )+(n-1) int cos^(n-2)x(1-cos^(2)x)dx`
`=(cos^(n-1) x sinx )+(n-1) int [cos^(n-2)x-cos^(n)x]dx`
or `I_(n)+(n-1) I_(n)=(cos^(n-1)x sinx)+(n-1)(I_(n-2))`
or `I_(n)=(1)/(n)(cos^(n-1) x sin x)+((n-1)/(n))I_(n-2)`

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