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Question Number 163158 by mnjuly1970 last updated on 04/Jan/22

p r o v e t h a t

\int_{0}^{\frac{π}{4}} \frac{s i n (x) + c o s (x)}{\sqrt{1 + s i n (x) c o s (x)}} d x = \sqrt{2} . {c o t}^{- 1} (\sqrt{2})

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Answered by mahdipoor last updated on 04/Jan/22

g e t s i n x - c o s x = \sqrt{3} u

{\begin{cases} \sqrt{3} d u = (c o s x + s i n x) d x \\ 3 u^{2} = {s i n}^{2} x + {c o s}^{2} x - 2 s i n x . c o s x \Rightarrow \frac{1 - 3 u^{2}}{2} = s i n x . c o s x \end{cases}

\Rightarrow \int_{0}^{π / 4} \frac{s i n x + c o s x}{\sqrt{1 + s i n x . c o s x}} d x = \int_{- 1 / \sqrt{3}}^{0} \frac{\sqrt{3} d u}{\sqrt{1 + \frac{1 - 3 u^{2}}{2}}} =

\sqrt{2} \int_{- 1 / \sqrt{3}}^{0} \frac{d u}{\sqrt{1 - u^{2}}} = {[\sqrt{2} {s i n}^{- 1} u + C]}_{- 1 / \sqrt{3}}^{0}

= - \sqrt{2} {s i n}^{- 1} (\frac{- 1}{\sqrt{3}}) = \sqrt{2} {s i n}^{- 1} (\frac{1}{\sqrt{3}})

i f s i n x = \frac{1}{\sqrt{3}} \Rightarrow c o t x = \sqrt{2}

\Rightarrow \sqrt{2} {s i n}^{- 1} (\frac{1}{\sqrt{3}}) = \sqrt{2} {c o t}^{- 1} (\sqrt{2})

Commented by mnjuly1970 last updated on 04/Jan/22

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