integration with limits (0 to pi/2) sin2x/sinx+cosx
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just convert everything tanx/2using sinx = 2tanx/2 / 1+ tan2 x/2 and cosx = (1- tan2 x/2) / (1+tan2 x/2)sin2x = 2sinx cosx then substitute tanx/2 = t . It will easily get resolved. Apply partial fractions after that.
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