`int (cos(x) + sin(x))/sin(2x) dx` Evaluate the integral

`int(cos x+sin x)/sin(2x) dx`

Let us denote the above integral wit `I.`

Use formula for sine of double angle: `sin2theta=2sin theta cos theta`

`int(cos x+sin x)/(2sin x cos x)dx=1/2int cos x/(sin x cos x)dx+1/2int sin x/(sin x cos x)dx=`

`1/2(int dx/sin x+int dx/cos x)`

Let us denote the above two integrals with `I_1` and `I_2` respectively.

To calculate `I_1` we use sine of double angle formula.

`I_1=int dx/(2sin(x/2) cos (x/2))`

Using the fact that `tan theta=sin theta/cos theta` we can write...

`int(cos x+sin x)/sin(2x) dx`

Let us denote the above integral wit `I.`

Use formula for sine of double angle: `sin2theta=2sin theta cos theta`

`int(cos x+sin x)/(2sin x cos x)dx=1/2int cos x/(sin x cos x)dx+1/2int sin x/(sin x cos x)dx=`

`1/2(int dx/sin x+int dx/cos x)`

Let us denote the above two integrals with `I_1` and `I_2` respectively.

To calculate `I_1` we use sine of double angle formula.

`I_1=int dx/(2sin(x/2) cos (x/2))`

Using the fact that `tan theta=sin theta/cos theta` we can write the above as

`int (1/2 cdot1/(cos^2(x/2)))/(tan(x/2))dx`

Now we use the fact that `int (f'(x))/(f(x))dx=ln|f(x)|+C.`

Therefore, since `(tan(x/2))'=1/2cdot1/(cos^2(x/2))` we have

`ln|tan(x/2)|+C`

Let us now calculate `I_2.`

`I_2=int dx/cosx`

Use the following formula: `cos theta=sin(theta+pi/2).`

`int dx/(sin(x+pi/2))`

Now we proceed as we did in calculating `I_2.` We use formula for sine of double angle.

`int dx/(2sin(x/2+pi/4)cos(x/2+pi/4))=int(1/2cdot1/(cos^2(x/2+pi/4)))/(tan(x/2+pi/4))dx`

Since `(tan(x/2+pi/4))'=1/2cdot 1/(cos^2(x/2+pi/4))` we get

`ln|tan(x/2+pi/4)|+C`

We can now calculate our starting integral `I.`

`I=1/2(I_1+I_2)=1/2(ln|tan(x/2)|+ln|tan(x/2+pi/4)|)+C`