`int_(pi/4)^(pi/3) csc^2(theta) d theta` Evaluate the integral

Textbook Question

Chapter 5, 5.4 - Problem 36 - Calculus: Early Transcendentals (7th Edition, James Stewart).
See all solutions for this textbook.

1 Answer | Add Yours

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to evaluate the definite integral using the fundamental theorem of calculus, such that:

`int_a^b f(u) du = F(b) - F(a)`

`int_(pi/4)^(pi/3) csc^2 theta d theta = int_(pi/4)^(pi/3) 1/(sin^2 theta) d theta = -cot theta|_(pi/4)^(pi/3)`

`int_(pi/4)^(pi/3) csc^2 theta d theta = -cot (pi/3) + cot (pi/4)`

`int_(pi/4)^(pi/3) csc^2 theta d theta = 1 - (sqrt3)/3 = (3 - sqrt3)/3`

Hence, evaluating the definite integral yields` int_(pi/4)^(pi/3) csc^2 theta d theta = (3 - sqrt3)/3.`

We’ve answered 318,928 questions. We can answer yours, too.

Ask a question