Question

($\sqrt{\cot x}$+ $\sqrt{\tan x}$) dx. equals:

• A. $\sqrt{2}$. sin^–1(sin x – cos x)
• B. $\sqrt{2}$sin^–1 (sin x + cos x)
• C. 2 sin^–1 (sin x – cos x)
• D. None of these

Answer 1

Answer: (A)

$\sqrt{2}$. sin^–1(sin x – cos x)

Answer 2

= $\int \left( \sqrt { \frac { \cos x } { \sin x } } + \sqrt { \frac { \sin x } { \cos x } } \right)$x

= dx

= $\sqrt { 2 }$ $\int_{}^{}\frac{\cos x + \sin x}{\sqrt{\sin 2x}}$dx

put t = sin x – cos x

dt = (cos x + sin x) dx

also t² = 1 –sin 2x

\ I = $\sqrt { 2 }$ $\int_{}^{}\frac{dt}{\sqrt{1 - t^{2}}}$

= $\sqrt { 2 }$sin^–1 t

= $\sqrt { 2 }$sin^–1 (sin x –cos x) + c