The integral value=?

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1 Answer
Mar 15, 2018

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