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