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If the value of the definite integral `int_0^1^(2007)C_7x^(2000)dot(1-x)^7dx` is equal to `1/k ,w h e r ek in N ,` then the value of `k/(26)` is____

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The correct Answer is:
208
Let `I=int_(0)^(1).^(207)C_(7).ubrace(x^(200))_(II).ubrace((1-x)^(7))_(I)dx`
`=.^(207)C_(7)[(1-x)^(7).(x^(201))/201|_(0)^(1)+7/201int_(0)^(1)(1-x)^(6).x^(201)dx]`
`=.^(207)C_(7) . 1/207 int_(0)^(1)(1-x)^(6) . x^(201)dx`
Integrating by parts againsix times more we get
`I=.^(207)C_(7).(7!)/(201.202.203.204.205.206.207)int_(0)^(1)x^(207)dx`
`=((207)!)/(7!(200)!) . (7!)/(201.202......207) . 1/208`
`=((207)!)/((207)!7!) . (7!)/208=1/208=1/k`
or `k=208`
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