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Evaluate int(0)^(pi//4)ln(1+tanx)dx...

Evaluate `int_(0)^(pi//4)ln(1+tanx)dx`

Text Solution

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Let `I=int_(0)^(pi//4)log(1+tanx)dx=int_(0)^(pi//4)log[1+(1-tanx)/(1+tanx)]dx=int_90)^(pi//4)log2dx-int_(0)^(pi//4)-log(1+tanx)dx`
`=(pi)/(4)log2-I`
So, `I=(pi)/(8)log2`
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