Question

`int_(pi/5)^((3pi)/10) [(tan x)/(tan x + cot x)]`dx = ?

Options

• `(3pi)/10`

• `pi/20`

• `pi/2`

• `pi/5`

Answer 1

`pi/5`

Explanation:

Let I = `int_(pi/5)^((3pi)/10) (tan x)/(tan x + cot x)`dx

I = `int_(pi/5)^((3pi)/10) (tan ((3pi)/10 + pi/5 - x))/(tan ((3pi)/10 + pi/5 - x) + cot ((3pi)/10 + pi/5 - x))`

I = `int_(pi/5)^((3pi)/10) (cot x)/(cot x + tan x)`dx

2I = `int_(pi/5)^((3pi)/10) ((tan x + cot x)/(tan x + cot x)) "dx" = int_(pi/5)^((3pi)/10)`dx

I = `1/2((3pi)/10 - pi/5) = pi/20`