The question is evaluate the integral of : dz using integration by parts. I have u=z,du=1,v=(-10^z)/ln(10),dv=10^z I am stuck....

Your idea is correct! But you got the integra wrong for v'(z) (you forgot a minus sign!). The correct one is:

You can check that this is the correct v(z) by taking its derivative with respect to z. You got it right with u(z) and u'(z), so I will use then for the integration by parts.

Now, following the integration by parts we have:

Inserting our functions:

All that remains is to evaluate the integral of 

But we know that integral. It is simply 

Thus, we get as a final result:

 

I used D for constant because that may not be the same value as C, due to the integral being indefinite on the first term aswell!

I believe your problem was with the minus sign, which could have resulted in a really complicated integral.