Problem in integrating to find Rutherford's formula

[MatinSAR]

Homework Statement

Figure below.

Relevant Equations

Figure below.

Could someone guide me on what change of variable was used to obtain equation 9.138 from equation 9.137?

Book : Classical Dynamics of Particles and Systems 5th Edition by Stephen T. Thornton (Author), Jerry B. Marion (Author)

They told us to check equation 8.38 and in that page they had $1/r^2$ in the numerator so they used $u=1/r$ then they get $du=-dr/r^2$.
But I cannot use that here because I have $1/r$ in the numerator ...

 

[Hill]

If you take $r$ out from the square root in the denominator you get $1/r^2$ in the numerator.

 

[MatinSAR]

If you take $r$ out from the square root in the denominator you get $1/r^2$ in the numerator.

I was oblivious. Thanks for your clever idea! I will retry to solve.