how to integrate √(cotx.cosecx)dx

Question

how to integrate √(cotx.cosecx) with respect to x

Answer 1

∫√cot(x)csc(x)dx

.

Draw a right triangle with the hypotenuse equal to 1.  Let one of the non-right angles equal x.

.

     |

     |_____x_    You need to imagine the hypotenuse....

.

The left side of the triangle has length sin(x)  ...  because the hypotenuse is 1.

.

We could let the bottom leg be cos(x) - but let's use √(1-sin²(x)

.

∫√cot(x)csc(x)dx = ∫√(1-sin²(x)/sin(x)  *1/sin(x) )dx

.

   = ∫√(1-sin²(x)/sin²(x) dx

.

Separate the radical into 2 different fractions

.

   ∫√[1/sin²(x) - sin²(x)/sin²(x)]   dx  = ∫√[csc²(x) - 1]   dx

.

hmmmm  that looks like an identity!

.

   = ∫√cot²(x)   dx   = ∫cot(x) dx = csc(x) + cot(x) + C