Evaluate the following integral: `int_0^(pi//2)(x+sinx)/(1+cosx)dx`

Explanation

Given,I=`int_0^(pi//2)(x+sinx)/(1+cosx)dx`

I=`int_0^(pi//2)(x-2sin(x/2)cos(x/2))/(2cos^(x/2))dx`

I=`1/2int_0^(pi//2)(xsec^2(x/2)dx+int_0^(pi//2)tan(x/2))dx`


I=`1/2[((xtan(x/2))/(1/2))-int_0^(pi//2) tan(x/2)/(1/2)dx]+int_0^(pi//2)tan(x/2)dx`


I=`(xtan(x/2))_0^(pi//2)-int_0^(pi/2) tan(x/2)dx+int_0^(pi//2)tan(x/2)dx`

`I=pi/2tan(pi/4)`

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