Question

Evaluate : `int_(π/6)^(π/3) 1/(1 + sqrt(cotx)` dx

Answer 1

Let I = `int_(π/6)^(π/3) 1/(1 + sqrt(cotx)` dx

      = `int_(π/6)^(π/3) 1/(1 + sqrt(cotx/sinx)` dx

      = `int_(π/6)^(π/3) sqrt(sinx)/(sqrt(sinx)+ sqrt(cotx)`......(1)

By property `int_a^b f(x) dx = int_a^b f(a + b - x) dx`

I = `int_(π/6)^(π/3) sqrt(sin (π/6 + π/3) - x)/sqrt(sin (π/6 + π/3) + sqrt(sin (π/6 + π/3)`

I = `int_(π/6)^(π/3) sqrt(sin (π/2- x))/sqrt(sin (π/2 - x) + sqrt(sin (π/2 - x)` dx

I = `int_(π/6)^(π/3) sqrt(cos x)/sqrt(cosx + sqrt(sinx)` dx........(2) 

Adding (1) and (2)

21 = `int_(π/6)^(π/3) (sqrt(sinx) + sqrt(cos x))/sqrt(sinx+ sqrt(sinx)` dx

21 = `[x]_(π//3)^(π//6)`

21 = `π/3 - π/6 = π/6`

21 = `π/6`

I = `π/12`