Is there any relationship to the Gaussian integral?

Calculus Level 4

The Gaussian integral states that $\displaystyle \int_{-\infty}^{+\infty} e^{-x^2 } \, dx = \sqrt{\pi }$ .

Given that information (if it is related to the problem at hand), evaluate

$\int_{- \infty}^{+ \infty} x^{2} e^{-x^4} \, dx$

If the integral above can be represented in the form

$\dfrac{a}{b} \Gamma \left( \dfrac{c}{d}\right) \; ,$

where $a,b,c$ and $d$ are positive integers such that $\gcd(a,b)=\gcd(c,d) =1$ , find $a+b+c+d$ .

The answer is 10.

2 solutions

Chew-Seong Cheong
Apr 4, 2016

$\begin{aligned} I & = \int_{-\infty}^\infty x^2 e^{-x^4} dx \quad \quad \small \color{#3D99F6}{\text{Since the function is even,}} \\ & = 2 \int_0^\infty x^2 e^{-x^4} dx \quad \quad \small \color{#3D99F6}{\text{Let }t = x^4 \quad \Rightarrow dx = \frac{1}{4}t^{-\frac{3}{4}} dt} \\ & = \frac{1}{2} \int_0^\infty t^{\frac{1}{2}-\frac{3}{4}} e^{-t} dt \\ & = \frac{1}{2} \int_0^\infty t^{-\frac{1}{2}} e^{-t} dt \\ & = \frac{1}{2} \Gamma \left(\frac{3}{4} \right) \end{aligned}$

$\Rightarrow a + b + c + d = 1+2+3+4 = \boxed{10}$

Vignesh S
Apr 3, 2016

put $x^4=t$ , and that's it the problem is over!!

Is there any relationship to the Gaussian integral?

The answer is 10.

2 solutions

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