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If I(n)=int(0)^(1)(1+x+x^(2)+....+x^(n-1...

If `I_(n)=int_(0)^(1)(1+x+x^(2)+....+x^(n-1))(1+3x+5x^(2)+....+(2n-3)x^(n-2)+(2n-1)x^(n-1))dx,n in N,` then the value of `sqrt(I_(9))` is

A

3

B

6

C

9

D

12

Text Solution

Verified by Experts

The correct Answer is:
C
Let `x=t^(2)`
`I_(n)=int_(0)^(1)2t(1+t^(2)+t^(4)+....+t^(2n-2))(1+3t^(2)+5t^(4)+.....+(2n-3)t^(2n-4)+(2n-1)t^(2n-n))dt`
`=int_(0)^(1)2(t+t^(3)+t^(5)+.....+t^(2n-1))(1+3t^(2)+5t^(4)+....+(2n-3)t^(2n-4)+(2n-1)t^(2n-2))dt`
Let `y=t+t^(3)+t^(5)+....+t^(2n-1)`
`therefore" "I_(n)=2int_(0)^(n)tdt=n^(2)`
`therefore" "sqrt(I_(9))=sqrt(9^(2))=9`
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