Calculus: Early Transcendentals, Chapter 5, 5.5, Section 5.5, Problem 58

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

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