MHT CET · Maths · Definite Integration
\(\int_0^{\frac{\pi}{4}}(\sqrt{\tan x}+\sqrt{\cot x}) \mathrm{d} x=\)
- A \(\sqrt{2} \pi\)
- B \(\frac{\pi}{2}\)
- C \(2 \pi\)
- D \(\frac{\pi}{\sqrt{2}}\)
Answer & Solution
Correct Answer
(D) \(\frac{\pi}{\sqrt{2}}\)
Step-by-step Solution
Detailed explanation
\( \int_0^{\frac{\pi}{4}}(\sqrt{\tan x}+\sqrt{\cot x}) \mathrm{d} x = \int_0^{\frac{\pi}{4}} \frac{\sin x + \cos x}{\sqrt{\sin x \cos x}} \mathrm{d} x \) \( = \int_0^{\frac{\pi}{4}} \frac{\sqrt{2}(\sin x + \cos x)}{\sqrt{2 \sin x \cos x}} \mathrm{d} x = \int_0^{\frac{\pi}{4}} \frac{\sqrt{2}(\sin x + \cos x)}{\sqrt{\sin(2x)}} \mathrm{d} x \)
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