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Question
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`
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Solution
Let I = `int (1)/(3 + 2sinx).dx`
Put `tan (x/2) = t`
∴ x = 2 tan–1 t
∴ dx = `(2t)/(1 + t^2) and sinx = (2t)/(1 + t^2)`
∴ I = `int (1)/(3 + 2((2t)/(1 + t^2))).(2dt)/(1 + t^2)`
= `int (1 + t^2)/(3 + 3t^2 + 4t).(2dt)/(1 + t^2)`
= `2 int (1)/(3t^2 + 4t + 3)dt`
= `(2)/(3) int (1)/(t^2 + 4/3t + 1)dt`
= `(2)/(3) int (1)/((t^2 + 4/3t + 4/9) - (4)/(9) + 1)dt`
= `(2)/(3) int (1)/((t + 2/3)^2 + (sqrt(5)/3)^2)dt`
= `(2)/(3) xx (1)/((sqrt(5)/3))tan^-1 [(t + 2/3)/(sqrt(5)/(3))] + c`
= `(2)/sqrt(5)tan^-1 ((3t + 2)/sqrt(5)) + c`
= `(2)/sqrt(5)tan^-1 [(3tan(x/2) + 2)/sqrt(5)] + c`.
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