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If the value of the definite integral `int_0^1(sin^(-1)sqrt(x))/(x^2-x+1)dx`is `(pi^2)/(sqrt(n))`(where `n in N),`then the value of `n/(27)`is |
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Answer» Correct Answer - 108 `I=int_(0)^(1)(sin^(-1)sqrt(x))/(x^(2)-x+1)dx`……………1 `I=int_(0)^(1)(sin^(-1)sqrt(1-x))/(x^(2)-x+1)dx=int_(0)^(1)(cos^(-1)sqrt(x))/(x^(2)-x+1)dx`…………..2 on adding 1 and 2 we get `2I=int_(0)^(1)(sin^(-1)sqrt(x)cos^(-1)sqrt(x))/(x^(2)-x+1) dx` `=(pi)/2 int_(0)^()(dx)/(x^(2)-x+1)dx` ` =(pi)/2 int_(0)^(1)(dx)((x-1/2)^(2)+((sqrt(3))/2)^(2))dx` `:.2I=(pi)/2 1/(((sqrt(3))/2))[tan^(-1)((2x-1)/(sqrt(3)))]_(0)^(1)=(pi)^(2)/(3sqrt(3))` Hence `I=(pi^(2))/(6sqrt(3))=(pi^(2))/(sqrt(108))=(pi^(2))/(sqrt(n))` |
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