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Whats the step

I wanted to express

\(\large{\displaystyle \int_{0}^{1} \frac{dx}{\sqrt{1+x^{4}}}}\) in terms of beta function

I started with \(x=\sqrt{\tan t}\), then got the integral in form of \(sin2t\) and again substituted \(sin2t=z\) , after this i couldn't solve

Note by Tanishq Varshney
2 years, 1 month ago

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I assume you got to this point: \( \displaystyle \frac 1{\sqrt2} \int_0^{\pi /4} (\sin(2t))^{-1/2} \, dt \),

Now let \(y = 2t \), you get \( \displaystyle \frac1{2\sqrt2} \int_0^{\pi /2} (\sin(y))^{-1/2} \, dy \).

Compare with equation 14 and equation 1 from this page.

And apply Euler Reflection Formula, you should get the answer of \( \frac1{8\sqrt\pi} \left [ \Gamma \left( \frac 14\right)^2 \right ] \). Pi Han Goh · 2 years, 1 month ago

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@Pi Han Goh Thanx for the help \(\ddot \smile\) Tanishq Varshney · 2 years, 1 month ago

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@Tanishq Varshney Sry couldn't make it in time . Btw are you participating in Proofathon ? Azhaghu Roopesh M · 2 years, 1 month ago

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@Azhaghu Roopesh M Nope, i am just a beginner to all this , not good as all other members on brilliant Tanishq Varshney · 2 years, 1 month ago

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@Tanishq Varshney No prob , just give it a try .Btw I am not able to solve 4 questions as of now . Azhaghu Roopesh M · 2 years, 1 month ago

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