<s>[INST] How to prove $\int_{0} ^{1}{\sqrt{1-x^4}dx}=\frac{1}{12}\sqrt {\frac{2}{\pi}}(\Gamma\left(\frac{1}{4}\right))^2$? To prove ,
$$\int_{0} ^{1}{\sqrt{1-x^4}dx}=\frac{1}{12}\sqrt {\frac{2}{\pi}}(\Gamma\left(\frac{1}{4}\right))^2$$
When we substitute $x^4$ with t
we get the equation
$$\frac{1}{4}\int_{0}^1t^{\frac{-3}{4}}(1-t)^{\frac{1}{2}}dt$$
the expression is in the forms of the beta functions, But I am not able to solve ahead. 
 [/INST] $$I=\frac { 1 }{ 4 } \int _{ 0 }^{ 1 } t^{ \frac { -3 }{ 4 }  }(1-t)^{ \frac { 1 }{ 2 }  }dt$$
Now, on using Beta function, we get $$I=\frac { 1 }{ 4 } B\left( \frac { 1 }{ 4 } ,\frac { 3 }{ 2 }  \right) $$
Now, we use the relation of Beta and gamma function. $$I=\frac { 1 }{ 4 } \frac { \Gamma \left( \frac { 1 }{ 4 }  \right) \Gamma \left( \frac { 3 }{ 2 }  \right)  }{ \Gamma \left( \frac { 7 }{ 4 }  \right)  } $$
Now, we use the formula: $\Gamma \left( n+1 \right) =n\Gamma \left( n \right) $
$$I=\frac { 1 }{ 4 } \frac { \Gamma \left( \frac { 1 }{ 4 }  \right) \Gamma \left( \frac { 1 }{ 2 }  \right) \times \frac { 1 }{ 2 }  }{ \frac { 3 }{ 4 } \Gamma \left( \frac { 3 }{ 4 }  \right)  } $$
We use this value: $\Gamma \left( \frac { 1 }{ 2 }  \right) =\sqrt { \pi  } $
$$I=\frac { \sqrt { \pi  }  }{ 6 } \frac { \Gamma \left( \frac { 1 }{ 4 }  \right)  }{ \Gamma \left( \frac { 3 }{ 4 }  \right)  } $$
$$I=\frac { \sqrt { \pi  }  }{ 6 } \frac { { \left( \Gamma \left( \frac { 1 }{ 4 }  \right)  \right)  }^{ 2 } }{ \Gamma \left( \frac { 1 }{ 4 }  \right) \Gamma \left( \frac { 3 }{ 4 }  \right)  } $$
Now, by Euler's Reflection Formula, we get $$I=\frac { \sqrt { \pi  }  }{ 6 } \frac { { \left( \Gamma \left( \frac { 1 }{ 4 }  \right)  \right)  }^{ 2 } }{ \frac { \pi  }{ \sin { \left( \frac { \pi  }{ 4 }  \right)  }  }  } $$
$$\boxed{\therefore \int _{ 0 }^{ 1 }{ \sqrt { 1-x^{ 4 } } dx } =\frac { 1 }{ 12 } \sqrt { \frac { 2 }{ \pi  }  } { \left( \Gamma \left( \frac { 1 }{ 4 }  \right)  \right)  }^{ 2 }}$$
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