A hydrogen atom is composed of one proton at the nucleus and one electron that moves around the nucleus. The electron does not move in a well-defined orbit but occupies a sphere shape "cloud" that surrounds the nucleus. At the lowest energy sate, the "cloud" can be assumed as a sphere centered at the nucleus. The way to describe the motion of the electron is to find the probability that the electron will be found within a certain area.

What is known:

The probability density function for the electron radial location is:

      p(r) = 4/a³r²e^-2x/a

where a is a constant. Its integral gives the probability that the electron will be found within a certain area:

      

where r is the radius of the sphere centered at the nucleus where the electron will be found and a is a constant.

Use integration by parts. Let:
      u = x²
      dv = e^-2x/a dx