$$ x^{2} + \sqrt{ \frac{ \pi }{2} } \Gamma ^{-2} (\frac{3}{4} ) \int _0^ \frac{ \pi }{2} \frac{xsinx}{ \sqrt{cosx} } dx=3x$$

Question

معادله زیر را در مجموعه اعداد حقیقی حل کنید: $$x^{2} + \sqrt{ \frac{ \pi }{2} } \Gamma ^{-2} (\frac{3}{4} ) \int _0^ \frac{ \pi }{2} \frac{xsinx}{ \sqrt{cosx} } dx=3x$$

Comments

$$I= \int _0^ \frac{ \pi }{2}  \frac{xsinx}{ \sqrt{cosx} } dx= \underbrace{-2x \sqrt{cosx} ]_0^ \frac{ \pi }{2} =0}  +2 \int _0^ \frac{ \pi }{2}  \sqrt{cosx} dx \wedge 2 \int _0^ \frac{ \pi }{2}  \sqrt{cosx} dx = \beta ( \frac{1}{2} , \frac{3}{4} )= \frac{ \Gamma( \frac{1}{2} )  \Gamma ( \frac{3}{4} )}{ \Gamma( \frac{5}{4}  )}$$$$ \sqrt{ \frac{ \pi }{2} }  \frac{1 }{  \Gamma ^{2}( \frac{3}{4} ) }  \frac{ \sqrt{ \pi } \Gamma ( \frac{3}{4} ) }{ \frac{1}{4}  \Gamma ( \frac{1}{4} )}=2 \sqrt{2}  \frac{ \pi }{ \Gamma ( \frac{3}{4} ) \Gamma ( \frac{1}{4} )}  \wedge  \Gamma ( \frac{3}{4} ) \Gamma ( \frac{1}{4} )= \Gamma (1- \frac{1}{4} ) \Gamma ( \frac{1}{4} )= \frac{ \pi }{sin( \frac{ \pi }{4} )} = \pi  \sqrt{2}  \Longrightarrow 2 \sqrt{2}  \frac{ \pi }{ \Gamma ( \frac{3}{4} ) \Gamma ( \frac{1}{4} )} = \frac{2 \sqrt{2}  \pi }{ \pi  \sqrt{2} } =2  \Longrightarrow  x^{2} +2x=3x$$

Answer 1

$$I= \int _0^ \frac{ \pi }{2} \frac{xsinx}{ \sqrt{cosx} } dx= \underbrace{-2x \sqrt{cosx} ]_0^ \frac{ \pi }{2} =0} +2 \int _0^ \frac{ \pi }{2} \sqrt{cosx} dx \wedge 2 \int _0^ \frac{ \pi }{2} \sqrt{cosx} dx = \beta ( \frac{1}{2} , \frac{3}{4} )= \frac{ \Gamma( \frac{1}{2} ) \Gamma ( \frac{3}{4} )}{ \Gamma( \frac{5}{4} )} \Longrightarrow 2 \int _0^ \frac{ \pi }{2} \sqrt{cosx} dx= \frac{ \sqrt{ \pi } \Gamma ( \frac{3}{4} ) }{ \frac{1}{4} \Gamma ( \frac{1}{4} )} \wedge \Gamma ( \frac{1}{2} )= \sqrt{ \pi } \wedge \Gamma ( \frac{5}{4} )== \Gamma (1+ \frac{1}{4} )= \frac{1}{4} \Gamma ( \frac{1}{4} )$$$$ \Longrightarrow \sqrt{ \frac{ \pi }{2} } \frac{1 }{ \Gamma ^{2}( \frac{3}{4} ) } \frac{ \sqrt{ \pi } \Gamma ( \frac{3}{4} ) }{ \frac{1}{4} \Gamma ( \frac{1}{4} )}=2 \sqrt{2} \frac{ \pi }{ \Gamma ( \frac{3}{4} ) \Gamma ( \frac{1}{4} )} \wedge \Gamma ( \frac{3}{4} ) \Gamma ( \frac{1}{4} )= \Gamma (1- \frac{1}{4} ) \Gamma ( \frac{1}{4} )= \frac{ \pi }{sin( \frac{ \pi }{4} )} = \pi \sqrt{2} \Longrightarrow 2 \sqrt{2} \frac{ \pi }{ \Gamma ( \frac{3}{4} ) \Gamma ( \frac{1}{4} )} \Longrightarrow \Gamma ( \frac{3}{4} ) \Gamma ( \frac{1}{4} )= \Gamma (1- \frac{1}{4} ) \Gamma ( \frac{1}{4} )= \frac{ \pi }{sin( \frac{ \pi }{4} )} = \pi \sqrt{2} \Longrightarrow2 \sqrt{2} \frac{ \pi }{ \Gamma ( \frac{3}{4} ) \Gamma ( \frac{1}{4} )} =\frac{2 \sqrt{2} \pi }{ \pi \sqrt{2} } =2 \Longrightarrow x^{2} +2x=3x \Longrightarrow x^{2} -3x+2=0 \Longrightarrow x=1 \vee 2$$