Question

The integral value=?

Answer 1

`2 ln (sin(x/2))+C`

Explanation:

`1+2cot x(csc x+cotx) = 1+2 cos x/sin x (1/sin x+cos x/sin x)`
`qquad = 1+2 (cos x(1+cos x))/sin^2 x = (sin^2x+2cosx+2cos^2x)/sin^2x`
`qquad = (1+2cos x+cos^2x)/sin^2 x = ((1+cos x)/sin x)^2`

So, in the interval `x in (0,pi/2)`, we have :

`sqrt{1+2cot x(csc x+cotx)} = {1+cos x}/sin x = {2 cos^2(x/2)}/{2sin(x/2)cos(x/2)} =cot(x/2)`

Thus, the integral is

`int sqrt{1+2cot x(csc x+cotx)} dx = int cot(x/2) dx = 2 int cot(x/2) d(x/2) = 2 ln (sin(x/2))+C`