Ignited Integrals 4

Calculus Level 4

$\large \int_0^\infty \sqrt[4]x e^{-\sqrt x} \, dx$

If the value of the integral above is in the form $\dfrac AB \sqrt\pi$ , where $A$ and $B$ are coprime positive integers, find $A+B$ .

1 solution

A Former Brilliant Member
Feb 12, 2016

Let $\int_{0}^{\infty }{x^{\frac{1}{4}}}e^{-\sqrt[]{x}}dx = I$

Putting , $\sqrt{x}=t$ or $x=t^{2}$ and $dx=2tdt$ in above equation , we get ,

$I=\int_{0}^{\infty }t^{\frac{1}{2}}e^{-t}2tdt=2\int_{0}^{\infty}t^{\frac{3}{2}}e^{-t}dt=2\Gamma (\frac{5}{2})$

And hence , by definition ,

$=2.\frac{3}{2}\Gamma (\frac{3}{2}) = 2.\frac{3}{2}.\frac{1}{2}\Gamma (\frac{1}{2})= \frac{3\sqrt{\pi }}{2}$

So , here , $A$ = 3 and $B$ = 2 so , $A+B = {\color{magenta} 5}$

most of all your problems beta is the key

Mardokay Mosazghi - 5 years, 4 months ago