Question

If I = $\int_{}^{}\left( \sqrt{\tan x} + \sqrt{\cot x} \right)$dx, then I equals:

• A. $\sqrt{2}$sin^–1 (sin x + cos x) + C
• B. $\sqrt{2}$cos^–1 (sin x –cos x) + C
• C. $\sqrt{2}$sin^–1 (sin x –cos x) + C
• D. $\sqrt{2}$cos^–1 (sin x + cos x) + C

Answer 1

Answer: (C)

$\sqrt{2}$sin^–1 (sin x –cos x) + C

Answer 2

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

Put sin x – cos x = t, so that (sin x + cos x) dx = dt and

1 – 2 sin

x cos x = t²

\ I = $\sqrt{2}$ $\int_{}^{}\frac{dt}{\sqrt{1 - t^{2}}}$= $\sqrt{2}$sin^–1 (t) + C

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