Question

integral from 1/18 to 1/6 of csc(3pit)cot(3pit)dt

Answer 1

1/18 to 1/6 of csc(3pit)cot(3pit)dt
1/18 to 1/6 of [1/sin(3 pi t) ] cos(3pit) / [sin (3 pi t)] dt
1/18 to 1/6 of cos(3pit) / [sin^2 (3 pi t)] dt

if y = sin (3 pi t) then dy = 3 pi cos(3 pi t) dt
so
1/18 to 1/6 in t and [1/y^2] dy/ (3 pi)
if t= 1/18 then y = sin pi/6 = 1/2
if t = 1/6 then y = sin pi/2 = 1
so
you really have the integral from y = 1/2 to y = 1 of 1/(3 pi) times y^-2 dy
which is
1/2 to y = 1 of 1/(3 pi) times -1/y
if y = 1, -1/(3 pi)
if y = 1/2 , +2/(3 pi)
so 3/(3 pi) =1/(3pi)
so agree