Indefinite Integration 1 Question 39
40. The value of the integral $\int _0^{\pi / 2} \frac{\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}} d x$ is
(a) $\pi / 4$
(b) $\pi / 2$
(c) $\pi$
(d) None of these
$(1983,1 M)$
Assertion and Reason
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Solution:
- Let $I=\int _0^{\pi / 2} \frac{\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}} d x$
$\Rightarrow \quad I=\int _0^{\pi / 2} \frac{\sqrt{\tan x}}{\sqrt{\cot x}+\sqrt{\tan x}} d x$
On adding Eqs. (i) and (ii), we get
$$ \begin{aligned} & 2 I=\int _0^{\pi / 2} 1 d x \\ \therefore \quad I & =\frac{\pi}{4} \end{aligned} $$
