Assuming that we know only two things about the turbulent boundary layer over a flat

plate. They are the friction coefficient $C_{f,x}=\frac{0\mathrm{.}059}{(R{e}_x{)}^{\frac{1}{5}}}$ and the following one$-$fifth$-$power

law approximation for the boundary layer profile shape:

$\frac{u}{U}=(\frac{y}{}{)}^{\frac{1}{5}}$ for $y,\frac{u}{U}=1$ for $y>1.$ Using the definitions of the Karman

integral equations, determine the variation with respect to $x$ of:

a$)$ The displacement thickness.

b$)$ The momentum thickness.

c$)$ The boundary layer thickness.