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Question

Integrate the function tan1(1sinx1+sinx) w.r.t dx.

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Solution

I=tan11sinx1+sinxdxn
=tan1 cos2x/2+sin2x/22cosx/2sinx/2cos2x/2+sin2x/2+2cosx/2sinx/2dx
=tan1(cosx/2+sinx/2)2(cosx/2+sinx)2dx
=tan1(cosx/2sinx/2cosx/2+sinx/2)dx
=tan1(tan(π4x2))dx
=π4x2dx
=[π4xx24].

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