Q1Mathematics I

The given definite integral is the well-known standard Gaussian integral over the positive real axis:

To solve this analytically, we employ the powerful properties of the Euler Gamma function, which is fundamentally defined for real numbers $n>0$ by the continuous integral $\Gamma (n)={\int }_0^{\infty }{e}^{-t}{t}^{n-1}dt$. We can systematically transform our given integral into this precise Gamma format using a straightforward algebraic substitution. Let us set $x^2=t$. Differentiating both sides with respect to $x$ gives $2xdx=dt$, which allows us to isolate $dx$ as $dx=\frac{dt}{2x}=\frac{dt}{2\sqrt{t}}$.

Substituting these variables and maintaining the limits of integration (since as $x\to 0,t\to 0$ and as $x\to \infty ,t\to \infty$), the integral transforms as follows:

This mathematical form perfectly matches the strict definition of the Gamma function where the exponent $(n-1)=-1/2$, which strictly implies that $n=1/2$. Thus, we can rewrite the integral in terms of the Gamma function:

From standard mathematical tables and proofs, the exact geometric value of $\Gamma \left(\frac{1}{2}\right)$ is definitively known to be $\sqrt{\pi }$. Substituting this known value yields the final answer:

Thus, the exact value of the integral is $\frac{\sqrt{\pi }}{2}$.