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Evaluate: 5050(int0^1(1-x^(50))^(100)dx)...

Evaluate: `5050(int_0^1(1-x^(50))^(100)dx)/(int_0^1(1-x^(50))^(101)dx)`

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Verified by Experts

The correct Answer is:
5051
`int_(0)^(1)(1-x^(50))^(101)dx`
`=[x(1-x^(50))^(101)]_(0)^(1)-int_(0)^(1)x.101(1-x^(50))^(100)(-50x^(49))dx`
`=0+5050 int_(0)^(1)x^(50)(1-x^(50))^(100)dx`
`=-5050int_(0)^(1)(1-x^(50)-1)(1-x^(50))^(100)dx`
`=-5050int_(0)^(1)(1-x^(50))^(101)dx+5050int_(0)^(1)(1-x^(50))^(100)dx`
`=5051int_(0)^(1)(1-x^(50))^(101)dx=5050int_(0)^(1)(1-x^(50))^(100) dx`
`implies 5050(int_(0)^(1)(1-x^(50))^(100)dx)/((1-x^(50))^(101)dx)=5051`
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