Substitution of a definite integral

[bobsmith76]

Homework Statement

The Attempt at a Solution

I don't see why the √v disappears in this step

I understand how they got the 20/3, because they integrated, but if they're integrating then I would think v^-1/2 would become (v^1/2)/(1/2)

 

[Curious3141]

Homework Statement

The Attempt at a Solution

I don't see why the √v disappears in this step

I understand how they got the 20/3, because they integrated, but if they're integrating then I would think v^-1/2 would become (v^-1/2)/(1/2)

The √v disappears because $dv = \frac{2}{3}v^{-\frac{1}{2}}du$ (rearranging the equation they gave for du). Substitute that for dv into the integrand and you'll see why. The 20/3 comes from 10*(2/3) during this substitution, NOT from integration (which hasn't occurred yet).