Ngô Quốc Anh

September 30, 2007

Lời giải 1 bài tích phân bội, 3

Filed under: Các Bài Tập Nhỏ, Giải Tích 5 — Ngô Quốc Anh @ 3:47

Prove that

\displaystyle\int_0^1\int_0^1\dfrac{dx dy}{\left(-\ln (xy)\right)^{\frac{3}{2}}}=\boxed{\sqrt\pi}.

Solution. Use the substitution

\displaystyle u=xy,v={\frac {x}{y}}

we have that Jacobian J=-\frac{1}{2v} and thus

\displaystyle I=-\int\limits _{0}^{1}{du} \int\limits _{u}^{\frac{1}{u}}{dv}\frac{1}{2v}\frac{1}{(-\ln{u})^\frac{3}{2}}=-\int \limits_{0}^{1}\frac{du}{(-\ln{u})^\frac{3}{2}}\int\limits _{u}^{\frac{1}{u}}\frac{dv}{2v}=-\int\limits _{0}^{1} \frac{du}{(-\ln{u})^\frac{3}{2}}\ln{u}=\int\limits _{0}^{1}\frac{du}{(-\ln{u})^\frac{1}{2}}

Then we use substitution t=-\ln u and:

\displaystyle I=\int _{0}^{1}\! \frac{du}{(-\ln{u})^\frac{1}{2}}=\int _{0}^{\infty}\! \frac{\exp{t}}{t^\frac{1}{2}}dt=\Gamma \left( \frac{1}{2} \right)=\sqrt{\pi}.

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