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Question
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
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Solution
Let I = `int 1/(3+2sinx + cosx) dx`
Put tan `x/2 = t` Then `dx = 2/(1+ t^2) dt`
`sinx = (2t)/(1+t^2) and cos x = (1- t^2)/(1+ t^2)`
`:. I = int (2dt"/" (1+t^2))/(3+2 ((2t)/(1+t^2))+((1-t^2)/(1+t^2)))`
`= 2int (dt"/"(1+t^2))/((3(1+t^2) + 4t + (1-t^2))/(1+t^2))`
= `2int (dt)/(2t^2 + 4t + 4) = int (dt)/((t+1)^2 + 1)`
`= tan^(-1) (t + 1) + c`
`= tan^(-1)[tan (x/2) + 1)] + c`
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