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The value of of 5050 (int(0)^(1)(1-x^(50...

The value of of `5050 (int_(0)^(1)(1-x^(50))^(100)dx)/(int_(0)^(1)(1-x^(50))^(101)dx)` is________.

Text Solution

Verified by Experts

The correct Answer is:
5051
Let `I_(2)=int_(0)^(1)(1-x^(50))^(101)dx`,
`=[(1-x^(50))^(101)*x]_(0)^(1)+int_(0)^(1)(1-x^(50))^(100)50*x^(49)*x dx`
[ using intergration by parts] `=0-int_(0)^(1)(50)(101)(1-x^(50))^(100)(-x^(50))dx`
`=-(50)(101)int_(0)^(1)(1-x^(50))^(101)dx` `+(50)(101)int_(0)^(1)(1-x^(50))^(100)dx=505I_(2)+5050I_(1)`
`:. I_(2)+5050I_(2)=5050I_(2)`
`rArr ((5050)I_(1))/(I_(2))=5051`
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