0-pi-2-cot-x-cot-x-tan-x-dx-

Question Number 83146 by 09658867628 last updated on 28/Feb/20

\underset{0}{\int^{π / 2}} \frac{\sqrt{\cot x}}{\sqrt{\cot x} + \sqrt{\tan x}} d x =

Answered by Kunal12588 last updated on 28/Feb/20

I = \int_{0}^{π / 2} \frac{\sqrt{c o t x}}{\sqrt{c o t x} + \sqrt{t a n x}} d x

\Rightarrow I = \int_{0}^{π / 2} \frac{\sqrt{t a n x}}{\sqrt{t a n x} + \sqrt{c o t x}} d x

\Rightarrow I = \frac{1}{2} \int_{0}^{π / 2} d x

\Rightarrow I = \frac{1}{2} \times \frac{π}{2} = \frac{π}{4}

\underset{0}{\int^{π / 2}} \frac{\sqrt{\cot x}}{\sqrt{\cot x} + \sqrt{\tan x}} d x = \frac{π}{4}

Answered by niroj last updated on 28/Feb/20

let, I = \int_{0}^{\frac{π}{2}} \frac{\sqrt{\cot x}}{\sqrt{\cot x} + \sqrt{\tan x}} dx \dots . . (i)

= \int_{0}^{\frac{π}{2}} \frac{\sqrt{\cot (\frac{π}{2} - x)}}{\sqrt{\cot (\frac{π}{2}} - x) + \sqrt{\tan (\frac{π}{2} - x)}} dx ∵ \int_{0}^{a} xdx = \int_{0}^{a} (a - x) dx

= \int_{0}^{\frac{π}{2}} \frac{\sqrt{\tan x}}{\sqrt{\tan x} + \sqrt{\cot x}} dx \dots \dots (ii)

added (i) & (ii)

2 I = \int_{0}^{\frac{π}{2}} (\frac{\sqrt{\cot x}}{\sqrt{\cot x} + \sqrt{\tan x}} + \frac{\sqrt{\tan x}}{\sqrt{\tan x} + \sqrt{\cot x}}) dx

2 I = \int_{0}^{\frac{π}{2}} (\frac{\sqrt{\cot x} + \sqrt{\tan x}}{\sqrt{\cot x} + \sqrt{\tan x}}) dx

2 I = \int_{0}^{\frac{π}{2}} dx

2 I = {[x]}_{0}^{\frac{π}{2}}

2 I = (\frac{π}{2} - 0)

2 I = \frac{π}{2} \Rightarrow I = \frac{π}{4} / / .

Commented by peter frank last updated on 28/Feb/20

t h a n k y o u b o t h