Question

Evaluate: `int_(pi/12)^((5pi)/12)dx/(1+sqrtcotx)`

Answer 1

Let I = `int_(pi/12)^((5pi)/12)dx/(1+sqrtcotx)`

I = `int_(pi/12)^((5pi)/12)dx/(1+sqrt(cosx)/sqrtsinx)`

= `int_(pi/12)^((5pi)/12)sqrtsinx/(sqrtsinx+sqrtcosx)dx`  ...(I)

Put: `int_a^bf(x)dx=int_a^b(a+b-x)dx`

I = `int_(pi/12)^((5pi)/12)(sqrtsin(pi/2-x))/(sqrtsin(pi/2-x)+sqrt(cos(pi/2-x)))dx`

= `int_(pi/12)^((5pi)/12)sqrtcosx/(sqrtcosx+sqrtsinx)dx`  ...(II)

Adding (I) and (II)

2I = `int_(pi/12)^((5pi)/12)(sqrtsinx/(sqrtsinx+sqrtcosx)+sqrtcosx/(sqrtcosx+sqrtsinx))dx`

2I = `int_(pi/12)^((5pi)/12)dx`

2I = `[x]_(pi/12)^((5pi)/12)`

2I = `(5pi)/12-pi/12`

2I = `(4pi)/12`

2I = `pi/3`

I = `pi/6`