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If I(n)=int(0)^(pi)x^(n)sinxdx, then fin...

If `I_(n)=int_(0)^(pi)x^(n)sinxdx`, then find the value of `I_(5)+20I_(3)`.

Text Solution

Verified by Experts

The correct Answer is:
`pi^(5)`
`I_(m)=int_(0)^(pi)x^(m) sin x dx`
`=[-x^(m)cosx]_(0)^(pi)+n int_(0)^(pi)x^(n-1)cos x dx`
`=pi^(n)+n[x^(n-1)sinx ]_(0)^(pi)-n(n-1)int_(0)^(pi)x^(n-2)sin x dx`
`implies I_(m)=pi^(n)+n.0-n(n-1)I_(n-2)`
Put `n=5`
`I_(5)=pi^(5)-20I_(3)`
`I_(5)+20I_(3)=pi^(5)`
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