Evaluate : (i) `int((1+cosx))/((1-cosx))dx` (ii) `int((1+sinx))/((1+cosx))dx`

Explanation

`int((1+cosx))/((a-cosx))dx=int(2cos^(2)(x//2))/(2sin^(2)(x//2))dx`
`=intcot^(2)((x)/(2))dx=int("cosec"^(2)(x)/(2)-1)dx`
`=int"cosec"^(2)(x)/(2)dx-intdx`
`=int"cosec"^(2)tdt-intdx," where"(x)/(2)=t anddx=2dt`
`=-2cot t-x+C=-2cot((x)/(2))-x+C`.
(ii) `int((1+sinx)/(1+cosx))dx=int(1)/((1+cosx))dx+int(sinx)/((1+cosx))dx`
`=int(1)/(2cos^(2)(x//2))dx+int(2sin(x//2)cos(x//2))/(2cos^(2)(x//2))dx`
`=(1)/(2)intsec^(2)((x)/(2))dx+int "tan"(x)/(2)dx`
`=intsec^(2)tdt+2inttantdt,"where"(x)/(2)=t`
`=tant-2log|cost|+C`
`=tan((x)/(2))-2log|{:cos((x)/(2)):}|+C`.