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Question
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
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Solution
Let I = `int (1)/(2 + cosx - sinx).dx`
Put `tan (x/2)` = t
∴ x 2 tan–1 t
∴ dx = `(2dt)/(1 + t^2) and sin x = (2t)/(1 + t^2), cosx = (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)`
= `2 int (1)/(t^2 - 2t + 3)dt`
= `2 int (1)/((t^2 - 2t + 1) + 2)dt`
= `2 int (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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