Chapter 4 - Integrals - 4.3 The Fundamental Theorem of Calculus - 4.3 Exercises - Page 327: 31

1

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: $F(b) - F(a)$ $F(\frac{\pi}{2})-F(\frac{\pi}{6})$ $-csc(\frac{\pi}{2}) - -csc(\frac{\pi}{6})$ $-\frac{1}{sin(\frac{\pi}{2})} + \frac{1}{sin(\frac{\pi}{6})}$ $-\frac{1}{1} + \frac{1}{\frac{1}{2}} = -1 + 2 = 1$