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

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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.`