Answer
$1 - \frac{1}{\sqrt{3}}$
Work Step by Step
Evaluate the Integral: $\int^{\pi/3}_{\pi/4}csc^2(θ)dθ $
Recall the 2nd part of the Fundamental Theorem of Calculus:
$∫^b_af(x)dx=F(b)−F(a)$
Find $F(x)$:
$F(x) =\int csc^2(x)dx$
$F(x) =-cot(x)$
Now Evaluate: $F(b) - F(a)$
$F(\frac{\pi}{3})-F(\frac{\pi}{4})$
$-cot(\frac{\pi}{3}) - -cot(\frac{\pi}{4})$
$-\frac{cos(\frac{\pi}{3})}{sin(\frac{\pi}{3})} + \frac{cos(\frac{\pi}{4})}{sin(\frac{\pi}{4})}$
$-\frac{\frac{1}{2}}{\frac{\sqrt3}{2}} + 1$
$1 - \frac{1}{\sqrt 3}$
