# 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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