Ngô Quốc Anh

September 17, 2007

Một số bài tập tích phân bội

Filed under: Các Bài Tập Nhỏ, Giải Tích 5 — Ngô Quốc Anh @ 6:02
  • Chứng minh 

\int_{0}^{1}\int_{0}^{1}\;\;\frac{x\cdot\ln(xy)}{1-x^{2}y^{2}}\;\;\textbf dx\;\;\textbf dy\;\;=\;\;\boxed{-\frac{1}{2}\,\zeta{(2)}}

\int_{0}^{1}\;\int_{0}^{1}\;\;\frac{x}{(1+x^{2}y^{2})\cdot\ln\,(xy)}\;\;\textbf dx\;\textbf dy\;\;=\;\;\boxed{\ln\left(\frac{\sqrt{\;2\pi^{3}\;}}{\Gamma^{2}\left(\frac{1}{4}\right)}\right)}

\int_0^1\;\int_0^1\;\;\dfrac{\textbf dx\;\textbf dy}{\left(-\,\ln\,(x\,y)\right)^{\frac{3}{2}}}\;\;=\;\;\boxed{\sqrt\pi}

  •  Tính

\int_{0}^{1}\;\int_{0}^{1}\;\;\frac{\textbf dx\;\;\textbf dy}{1+x^{2}+y^{2}}


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