vertical wind gradients in the atmospheric boundary layer I'm reading the following paper:

Intercomparison of Bulk Aerodynamic Algorithms for the Computation of
  Sea Surface Fluxes Using TOGA COARE and TAO Data

and am having trouble working through some of the equations, I have copied these equations below:

Would someone be able to show me the process of how integrating (1) to (6) gives (7)?
Any help would be appreciated. 
 A: Just to clarify the text:


*

*Integrating (5) gives (7) (the very unstable conditions)

*Integrating (3) gives (8) (moderately unstable to marginally unstable)   

*Integrating (2) gives (9) (marginally to moderately stable) 

*Integrating (6) gives (10) (very stable). 


Equations (1) and (4) are definitions that are used. So for example, to get (9):
$$\phi_m = \frac{k z}{u_*}\frac{du}{dz} = 1 + 5\zeta = 1 + 5\frac{z}{L}$$
which is setting the definition given by (1) to the empirical correlation given by (6) and substituting in (4) for $\zeta$. Rearranging gives:
$$\frac{du}{dz} = \frac{u_*}{kz} + 5 \frac{u_*}{kL}$$
Integrating from $z_0$ to $\xi$ (a dummy variable of integration that will take the value $z$ when done):
$$u(z) = \frac{u_*}{k}\ln \xi |^{z}_{z_0} + 5 \frac{u_* \xi}{L} |^{z}_{z_0}$$
which simplifies to:
$$u(z) = \frac{u_*}{k}\left(\ln \frac{z}{z_0} + 5 \frac{z}{L}\right)$$
Then you would use (4) to convert that last term back into $\zeta$ which gives (9). 
A similar process is used for the rest.
