Question

How do you evaluate the definite integral `int 4/sin^2(z) dz` from `[pi/6, pi/2]`?

Answer 1

`4sqrt3`.

Explanation:

`int_(pi/6)^(pi/2) 4/sin^2 zdz=4int_(pi/6)^(pi/2)csc^2 zdz`

`=-4[cot z]_(pi/6)^(pi/2)`

`=-4[cot (pi/2)-cot (pi/6)]`

`=-4[0-sqrt3]`

`=4sqrt3`.