Question

Find the value of `int_0^1 tan^-1 ((1 - 2x)/(1 + x - x^2)) dx`.

Answer 1

`int_0^1 tan^-1 ((1 - 2x)/(1 + x - x^2)) dx`

= `int_0^1 tan^-1 (((1 - x) - x)/(1 + x(1 - x))) dx`

= `int_0^1 [tan^-1 (1 - x) - tan^-1x] dx`   

= `int_0^1 tan (1 - (1 - x))dx - int_0^1 tan^-1x  dx`...`["Because" int_0^a f(x)dx = int_0^a f(a - x)dx]`

= `int_0^1 tan^-1x  dx - int_0^1 tan^-1 x  dx = 0`

∴ `int_0^1 tan^-1 ((1 - 2x)/(1 + x - x^2)) dx = 0`