ISI2015-MMA-76

Question

Given that $\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$, the value of $$ \int_{-\infty}^{\infty}  \int_{-\infty}^{\infty} e^{-(x^2+xy+y^2)} dxdy$$ is

• A. $\sqrt{\pi/3}$
• B. $\pi/\sqrt{3}$
• C. $\sqrt{2 \pi/3}$
• D. $2 \pi / \sqrt{3}$

Comments

Option D is the correct answer

Answer 1

Hint: $x^2+xy+y^2=\frac{3x^2}{4}+(\frac{x}{2}+y)^2$

$\frac{x}{2}+y=t\Rightarrow dy=dt$