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The value of int(0)^(1)(tan^(-1)x)/(cot^...

The value of `int_(0)^(1)(tan^(-1)x)/(cot^(-1)(1-x+x^(2))dx` is____.

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The correct Answer is:
0.5
`I=int_(0)^(1)(tan^(-1)x)/(cot^(-1)(1-x+x^(2)))dx`
`=int_(0)^(1)("tan"^(-1)x)/("tan"^(-1)((1-x)+x)/(1-(1-x)x)) dx`
`=int_(0)^(1)(tan^(-1)x)/(tan^(-1)x+tan^(-1)(1-x))dx`…………..1
`=int_(0)^(1)(tan^(-1)(1-x))/(tan^(-1)(1-x)+tan^(-1)x)` …………..2
Adding 1 and 2 we get
`2I=int_(0)^(1)1dx=1`
`implies I=1/2`
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