Question

Evaluate the following integrals using properties of integration:

`int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan x))  "d"x`

Answer 1

Let I = `int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan x))  "d"x`   .......(1)

`int_"a"^"b" f(x)  "d"x = int_"a"^"b" f("a" + "b" - x)  "d"x`

I = `int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan (pi/8 + (3pi)/8 - x)))  "d"x`

= `int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan(pi/2 - x)))  "d"x`

= `int_(pi/8)^((3pi)/8) 1/(1 + sqrt(cot x))  "d"x`

= `int_(pi/8)^((3pi)/8) 1/(1 + 1/sqrt(tan x))  "d"x`

I = `int_(pi/8)^((3pi)/8) sqrt(tanx)/(1 + sqrt(tan x))  "d"x`  ........(2)

Add (1) + (2)

2I = `int_(pi/8)^((3pi)/8) (1 + sqrt(tan x))/(1 + sqrt(tan x))  "d"x`

= `int_(pi/8)^((3pi)/8)  "d"x`

= `[x]_(pi/8)^((3pi)/8)`

= `(3pi)/8- pi/8`

2I = `(2pi)/8 = (pi/4)` 

I = `pi/8`

`int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan x))  "d"x = pi/8`