int_(0)^(1)(log(1+n))/(1+n^(2))dn=(pi)/(8)log2

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int_(0)^(1)(log|1+x|)/(1+x^(2))dx=(pi)/(8)log2

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The value of int_(0)^(1)(8log(1+x))/(1+x^(2))dx is: (1)pi log2 (2) (pi)/(8)log2(3)(pi)/(2)log2(4)log2