Prove that integral of 0 to pie √x e^-x3 dx = √pie/3
Indefinite integral
"\\int \\sqrt{x}e^{-x^3}dx"Substitution "u=x^{3\/2}, du=\\dfrac{3}{2}x^{1\/2} dx"
"=\\dfrac{2}{3}\\int e^{-u^2}du"
Use that "\\displaystyle\\int_{0}^{\\infin}e^{-t^2}dt=\\dfrac{\\sqrt{\\pi }}{2}"
Then
"=\\dfrac{2}{3}(\\dfrac{\\sqrt{\\pi }}{2})=\\dfrac{\\sqrt{\\pi }}{3}"
Therefore
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