Question

Evaluate:

`int_(π/6)^(π/3) (root(3)(sinx))/(root(3)(sinx) + root(3)(cosx))dx`

Answer 1

Let I = `int_(π/6)^(π/3) (root(3)(sinx))/(root(3)(sinx) + root(3)(cosx))dx`  ...(1)

= `int_(π/6)^(π/3) root(3)(sin(π/2 - x))/(root(3)(sin(π/2 - x)) + root(3)(cos(π/2 - x)))dx`   ...`(∵ int_a^b f(x)dx = int_a^b f(a + b - x)dx)`

∴ I = `int_(π/6)^(π/3) root(3)(cosx)/(root(3)(cosx) + root(3)(sinx))dx`  ...(2)

Adding (1) and (2)

2I = `int_(π/6)^(π/3) (root(3)(sinx) + root(3)(cosx))/(root(3)(sinx) + root(3)(cosx))dx`

= `int_(π/6)^(π/3) 1 dx = [x]_(π/6)^(π/3)`

= `π/3 - π/6`

= `(2π - π)/6`

= `π/6`

∴ I = `π/12`