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Let I(n)=int(0)^(1)x^(n)sqrt(1-x^(2))dx....

Let `I_(n)=int_(0)^(1)x^(n)sqrt(1-x^(2))dx.` Then `lim_(nrarroo)(I_(n))/(I_(n-2))=`

A

2

B

1

C

`-1`

D

`-2`

Text Solution

Verified by Experts

The correct Answer is:
B
`I_(n)=int_(0)^(1)x^(n)sqrt(1-x^(2))dx=int_(0)^(1)x^(n-1)xsqrt(1-x^(2))dx`
`=-x^(n-1)|((1-x^(2))^(3//2))/(3)|_(0)^(1)+int_(0)^(1)(n-1)x^(n-1)((1-x^(2))^(3//2))/(3)dx`
`=0+(n-1)/(3)int_(0)^(1)x^(n-2)(1-x^(2))sqrt(1-x^(2))dx`
`=(n-1)/(3)I_(n-2)-(n-1)/(3)I_(n)`
`rArr" "3I_(n)+(n-1)I_(n)=(n-1)I_(n-2)`
`rArr" "(I_(n))/(I_(n)-2)=(n-1)/(n+2)`
`rArr" "underset(n rarroo)(lim)(I_(n))/(I_(n-2))=1.`
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