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The value of int0 1tan^(-1)((2x-1)/(1+x-...

The value of `int0 1tan^(-1)((2x-1)/(1+x-x^2))dx` is (A) 1 (B) 0 (C) -1 (D) `pi/4`

Text Solution

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`" Let "I= int_(0)^(1) tan^(-1) ((2x-1)/(1+x-x^(2)))dx`
` =int_(0)^(1) tan^(-1) [(x+(x-1))/(1-x(x-1))]dx`
`=int_(0)^(1) {tan^(-1) x+ tan^(-1) (x-1)}dx" "....(1)`
`rArr I = int_(0)^(1) {tan^(-1) x tan^(-1) (1-x)}dx`
`rArr I=int_(0)^(1)[tan^(-1) (1-x) -tan^(-1) {1-1-x)}"]"dx`
`rArr I = int_(0)^(1)[tan^(-1) (1-x)-tan^(-1) (x) ]dx " "...(2)`
Adding equations (1) and (2)
`2I =0 " " rArr " " I=0`
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