Answer
1
Work Step by Step
Evaluate the Integral: $\int^{\pi/2}_{\pi/6}csc(t)cot(t)dt$
Recall the 2nd part of the Fundamental Theorem of Calculus:
$∫^b_af(x)dx=F(b)−F(a)$
Find $F(x)$:
$F(x) =\int csc(x)cot(x)dx$
$F(x) =-csc(x)$
Now Evaluate: $F(b) - F(a)$
$F(\frac{\pi}{2})-F(\frac{\pi}{6})$
$-csc(\frac{\pi}{2}) - -csc(\frac{\pi}{6})$
$-\frac{1}{sin(\frac{\pi}{2})} + \frac{1}{sin(\frac{\pi}{6})}$
$-\frac{1}{1} + \frac{1}{\frac{1}{2}} = -1 + 2 = 1$