Question

Integration in square root of 1+cosx ÷ 1-cosx?

Answer 1

`int sqrt((1+cosx)/(1-cosx))dx = abs(cot(x/2)) /cot(x/2) ln abs sin(x/2)+C`

Explanation:

Use the trigonometric identities:

`1-cosx = 2sin^2(x/2)`

`1+cosx = 2cos^2(x/2)`

Then:

`int sqrt((1+cosx)/(1-cosx))dx = int sqrt (cos^2(x/2)/sin^2(x/2))dx = int abs cot(x/2)dx = 2int abs (cot t) dt`

So, where the cotangent is positive:

`int sqrt((1+cosx)/(1-cosx))dx = int cos(t)/sint dt = int (d(sint))/sint = ln abs sin(x/2)+C`

and in general:

`int sqrt((1+cosx)/(1-cosx))dx = abs(cot(x/2)) /cot(x/2) ln abs sin(x/2)+C`