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Question
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
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Solution
Let I = `int cot^-1 ((1 + sinx)/cosx)*dx`
`(1 + sinx)/cosx = (1 + cos(pi/2 - x))/(sin(pi/2 - x)`
= `(2cos^2(pi/4 - x/2))/(2sin(pi/4 - x/2)*cos(pi/4 - x/2)`
= `cot(pi/6- x/2)`
∴ I = `int cot^-1 [cot(pi/4 - x/2)]*dx`
= `int (pi/4 - x/2)*dx`
= `pi/(4) int 1*dx - 1/2 int x*dx`
= `pi/(4)*x - (1)/(2)*x^2/(2) + c`
= `pi/(4)x - (1)/(4)x^2 + c`.
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