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Evaluate int(pi//6)^(pi//3)(dx)/(1+tan^(...

Evaluate `int_(pi//6)^(pi//3)(dx)/(1+tan^(n)x)`

Text Solution

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Let `I=int_(pi//6)^(pi//3)(dx)/(1+tan^(n)x)=int_(pi//6)^(pi//3)(cos^(n)x)/(cos^(n)x+sin^(n)x)dx`………`(i)`
Also, `I=int_(pi//6)^(pi//3)(cos^(n)((pi)/(2)-x))/(cos^(n)((pi)/(2)-x)+sin^(n)((pi)/(2)-x))dx=int_(pi//6)^(pi//3)(sin^(n)x)/(sin^(n)x+cos^(n)x)dx`.........`(ii)`
From `(i) + (ii)`, we have `2I=int_(pi//6)^(pi//3)dx=(pi)/(6)`
Hence `I=(pi)/(12)`.
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