`int_(1/6)^(1/2) csc(pi t) cot(pi t) dt` Evaluate the definite integral.

Given `int_(1/6)^(1/2)csc(pit)cot(pit)dt`

Integrate using the Substitution Rule.

Let `u=pit`

`(du)/dt=pi`

`dt=(du)/pi`

`=int_(1/6)^(1/2)csc(u)cot(u)*(du)/pi`

`=1/piint_(1/6)^(1/2)csc(u)cot(u)du`

`=1/pi*[-csc(u)]`  Evaluated from t=1/6 to t=1/2

`=-1/picsc(u)` Evaluated from t=1/6 to t=1/2

Right now the limits of integration are in terms of t. Change the limits of integration to terms of u. 

Since `u=pit`

When `t=1/6` , `u=pi/6`

When  `t=1/2` , `u=pi/2`

`=-1/picsc(u)`  Evaluated from `u=pi/6`  to `u=pi/2`

`=1/(-pi)[csc(pi/2)-csc(pi/6)]`

`=1/-pi[1-2] `

`=1/-pi[-1]`

`=1/pi`