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Evaluate: int tan^(-1) {sqrt((1-sinx)/(1...

Evaluate: `int tan^(-1)` `{sqrt((1-sinx)/(1+sinx))}\ dx ,\ -pi//2

Text Solution

Verified by Experts

The correct Answer is:
`(pix)/(4) - (x^(2))/(4) + C`
`I = int tan^(-1) sqrt((1-cos(pi/2 - x))/(1+ cos(pi/2-x))) dx`
`= inttan^(-1) sqrt((2sin^(2)(pi/4-x/2))/(2cos^(2)(pi/4-x/2)))dx = inttan^(-1){tan(pi/4 - x/2)} dx`
`= int (pi/4- x/2) dx = (pix)/(4) - (x^(2))/(4) + C`.
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