Tag Archives: Dogbone

Solving an integral using Dogbone contour

Prove that $$\int^{1}_{0} \sqrt{x}\sqrt{1-x}\,dx = \frac{\pi}{8}$$ $$\textit{proof}$$ Consider the function $$f(z) = \sqrt{z-z^2} = e^{\frac{1}{2}\log(z-z^2)}$$ Consider the branch cut on the x-axis $$x(1-x)\geq 0\,\, \implies \, 0\leq x \leq 1 $$ Consider \( w= z-z^2 \) then $$\log(w) = \log|w|+i\theta,\,\, … Continue reading

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