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Question
Evaluate: `int (dx)/(2 + cos x - sin x)`
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Solution
Let I = `int (dx)/(2 + cos x - sin x)`
Put `tan x/2` = t
⇒ x = 2 tan–1t
∴ dx = `(2 dt)/(1 + t^2)`
And sin x = `(2t)/(1 + t^2)`, cos x = `(1 - t^2)/(1 + t^2)`
∴ I = `int 1/(2 + ((1 - t^2)/(1 + t^2)) - ((2t)/(1 + t^2))) * (2dt)/(1 + t^2)`
= `int (1 + t^2)/(2 + 2t^2 + 1 - t^2 - 2t) * (2dt)/(1 + t^2)`
= `2int 1/(t^2 - 2t + 3) dt`
= `2int 1/((t - 1)^2 + (sqrt(2))^2) dt`
= `2 xx 1/sqrt(2) tan^-1 ((t - 1)/sqrt(2)) + C`
= `sqrt(2)tan^-1 ((tan(x/2) - 1)/sqrt(2)) + C`
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