Question

need quality Answer

Integrate :-

${\displaystyle {\int \dfrac{1}{\sqrt{x} \sqrt{1-x} }}}$



spam=strict answer will be taken .​

Answer 1

Step-by-step explanation:

$[tex]{\displaystyle {\int \dfrac{1}{\sqrt{x} \sqrt{1-x} }}}$

[/tex]

$We have</p><p></p><p>\displaystyle \bf \int \dfrac{ \sqrt{x^{2} + a^{2} } }{x} dx∫xx2+a2dx</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>\implies\displaystyle \sf \int \dfrac{ \sqrt{x^{2} + a^{2} } \sqrt{x^{2} + a ^{2} } }{x \sqrt{x {}^{2} + a ^{2} } } dx⟹∫xx2+a2x2+a2x2+a2dx</p><p></p><p>⠀⠀⠀</p><p></p><p>\implies \displaystyle \sf \int \dfrac{x^{2} + {a}^{2} }{x \sqrt{ {x}^{2}+{a}^{2}} } dx⟹∫xx2+a2x2+a2dx</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>\implies \displaystyle \sf \int \dfrac{x^{2} dx}{x \sqrt{ {x}^{2} + {a}^{2} } } + \int\dfrac{a^{2} dx}{x \sqrt{ {x}^{2} + {a}^{2} }}⟹∫xx2+a2x2dx+∫xx2+a2a2dx</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>\implies \displaystyle \sf \dfrac{1}{2} \int(2x )( {x}^{2} + {a}^{2} )^{ - \frac{1 }{2} } dx + {a}^{2} \int \frac{dx}{x \sqrt{ {x}^{2} + {a}^{2} } }⟹21∫(2x)(x2+a2)−21dx+a2∫xx2+a2dx</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>\implies \displaystyle \sf \frac{1}{2} \bigg \{ \sf \dfrac{( {x}^{2} + {a}^{2} ) ^{ \frac{1}{2} } }{ \dfrac{1}{2} } \bigg \} + a^{2} I_{1} + c....(1)⟹21{21(x2+a2)21}+a2I1+c....(1)</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>Where</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>\displaystyle \sf I_{1} = \int \dfrac{dx}{x \sqrt{ {x}^{2} + {a}^{2} } }I1=∫xx2+a2dx</p><p></p><p>✶Put x=1/t ⇛dx=-1 /t² dt</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>\implies \displaystyle \sf \int \frac{ - \dfrac{1}{{t}^{2}}dt }{ \dfrac{1}{t} \sqrt{ \dfrac{1 }{ {t}^{2}} + {a}^{2} } }⟹∫t1t21+a2−t21dt</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>\implies \displaystyle \sf \int \dfrac{ - dt}{ \sqrt{1 + {a}^{2} {t}^{2} } }⟹∫1+a2t2−dt</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>\implies \displaystyle \sf \frac{1}{a} \int \dfrac{dt}{ \sqrt{ {t}^{2} + \dfrac{1}{ {a}^{2} } } }⟹a1∫t2+a21dt</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>\implies \displaystyle \sf - \frac{1}{a} log \bigg(t + \sqrt{ {t}^{2} + \dfrac{1}{ {a}^{2} } } \bigg)⟹−a1log(t+t2+a21)</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>\implies \displaystyle \sf - \frac{1}{a} log \bigg( \frac{1}{x} + \sqrt{ \frac{1}{ {x}^{2} } + \dfrac{1}{ {a}^{2} } } \bigg)⟹−a1log(x1+x21+a21)</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>\implies \displaystyle \sf - \frac{1}{a} log \bigg( \frac{a + \sqrt{ {x}^{2} + {a}^{2} } }{ax} \bigg)....(2)⟹−a1log(axa+x2+a2)....(2)</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>$

Answer 2

Step-by-step explanation:

$[tex]∫x1−x1</p><p></p><p>$

We have \displaystyle \bf \int \dfrac{ \sqrt{x^{2} + a^{2} } }{x} dx∫xx2+a2dx \implies\displaystyle \sf \int \dfrac{ \sqrt{x^{2} + a^{2} } \sqrt{x^{2} + a ^{2} } }{x \sqrt{x {}^{2} + a ^{2} } } dx⟹∫xx2+a2x2+a2x2+a2dx <\implies \displaystyle \sf \int \dfrac{x^{2} + {a}^{2} }{x \sqrt{ {x}^{2}+{a}^{2}} } dx⟹∫xx2+a2x2+a2dx\implies \displaystyle \sf \int \dfrac{x^{2} dx}{x \sqrt{ {x}^{2} + {a}^{2} } } + \int\dfrac{a^{2} dx}{x \sqrt{ {x}^{2} + {a}^{2} }}⟹∫xx2+a2x2dx+∫xx2+a2a2dx \implies \displaystyle \sf \dfrac{1}{2} \int(2x )( {x}^{2} + {a}^{2} )^{ - \frac{1 }{2} } dx + {a}^{2} \int \frac{dx}{x \sqrt{ {x}^{2} + {a}^{2} } }⟹21∫(2x)(x2+a2)−21dx+a2∫xx2+a2dx\implies \displaystyle \sf \frac{1}{2} \bigg \{ \sf \dfrac{( {x}^{2} + {a}^{2} ) ^{ \frac{1}{2} } }{ \dfrac{1}{2} } \bigg \} + a^{2} I_{1} + c....(1)⟹21{21(x2+a2)21}+a2I1+c....(1) < /p > < p > < /p > < p > ⠀⠀⠀⠀ < /p > < p > < /p > < p > Where < /p > < p > < /p > < p > ⠀⠀⠀⠀ < /p > < p > < /p > < p > \displaystyle \sf I_{1} = \int \dfrac{dx}{x \sqrt{ {x}^{2} + {a}^{2} } }I1=∫xx2+a2dx < /p > < p > < /p > < p > ✶Put x=1/t ⇛dx=-1 /t² dt < /p > < p > < /p > < p > ⠀⠀⠀⠀ < /p > < p > < /p > < p > \implies \displaystyle \sf \int \frac{ - \dfrac{1}{{t}^{2}}dt }{ \dfrac{1}{t} \sqrt{ \dfrac{1 }{ {t}^{2}} + {a}^{2} } }⟹∫t1t21+a2−t21dt < /p > < p > < /p > < p > ⠀⠀⠀⠀ < /p > < p > < /p > < p > \implies \displaystyle \sf \int \dfrac{ - dt}{ \sqrt{1 + {a}^{2} {t}^{2} } }⟹∫1+a2t2−dt < /p > < p > < /p > < p > ⠀⠀⠀⠀ < /p > < p > < /p > < p > \implies \displaystyle \sf \frac{1}{a} \int \dfrac{dt}{ \sqrt{ {t}^{2} + \dfrac{1}{ {a}^{2} } } }⟹a1∫t2+a21dt < /p > < p > < /p > < p > ⠀⠀⠀⠀ < /p > < p > < /p > < p > \implies \displaystyle \sf - \frac{1}{a} log \bigg(t + \sqrt{ {t}^{2} + \dfrac{1}{ {a}^{2} } } \bigg)⟹−a1log(t+t2+a21) < /p > < p > < /p > < p > ⠀⠀⠀⠀ < /p > < p > < /p > < p > \implies \displaystyle \sf - \frac{1}{a} log \bigg( \frac{1}{x} + \sqrt{ \frac{1}{ {x}^{2} } + \dfrac{1}{ {a}^{2} } } \bigg)⟹−a1log(x1+x21+a21) < /p > < p > < /p > < p > ⠀⠀⠀⠀ < /p > < p > < /p > < p > \implies \displaystyle \sf - \frac{1}{a} log \bigg( \frac{a + \sqrt{ {x}^{2} + {a}^{2} } }{ax} \bigg)....(2)⟹−a1log(axa+x2+a2)....(2) < /p > < p > < /p > < p > ⠀⠀⠀⠀ < /p > < p > < /p > < p >Wehave</p><p></p><p>∫xx2+a2dx∫xx2+a2dx⟹∫xx2+a2x2+a2x2+a2dx⟹∫xx2+a2x2+a2x2+a2dx</⟹∫xx2+a2x2+a2dx⟹∫xx2+a2x2+a2dx⟹∫xx2+a2x2dx+∫xx2+a2a2dx⟹∫xx2+a2x2dx+∫xx2+a2a2dx⟹21∫(2x)(x2+a2)−21dx+a2∫xx2+a2dx⟹21∫(2x)(x2+a2)−21dx+a2∫xx2+a2dx</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>⟹21{21(x2+a2)21}+a2I1+c....(1)⟹2121(x2+a2)21+a2I1+c....(1)</p><p></p><p>⠀⠀⠀⠀</p><p></p><p>Where</I1=∫xx2+a2

[/tex]