Question

The value of $\displaystyle \int_{0}^{\infty} e^{-x^2} \, dx$ is

• A. $\dfrac{1}{3}\Gamma\left(\dfrac{3}{2}\right)$
• B. $\dfrac{1}{2}\Gamma\left(\dfrac{1}{2}\right)$
• C. $\dfrac{1}{2}\Gamma\left(\dfrac{3}{2}\right)$
• D. $\Gamma\left(\dfrac{1}{5}\right)$

Hint:

Use the identity $\int_0^\infty e^{-x^2} dx = \dfrac{\sqrt{\pi}}{2}$ and link it with $\Gamma(1/2) = \sqrt{\pi}$.

Answer 1

The Correct Option is B

Step 1: Recognize standard Gaussian integral
\[ \int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2} \] Step 2: Recall Gamma function relation
\[ \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi} \Rightarrow \frac{1}{2} \Gamma\left(\frac{1}{2}\right) = \frac{\sqrt{\pi}}{2} \] Hence, \[ \boxed{\int_0^\infty e^{-x^2} dx = \frac{1}{2} \Gamma\left(\frac{1}{2}\right)} \]