Why Is My Integration Approach Not Working?

[Chewy0087]

Homework Statement

http://i27.photobucket.com/albums/c171/Chewbacc0r/problem.jpg

The Attempt at a Solution

Basically, for the life of me I can't see how they get from "This expression may be written as..." to "Which may be integrated to contain..."

Now, I have no problem with the integration at all, I just can't see how they've put it in the right form.

Surely;

$$\frac{1}{v_{0}}+\frac{Ct}{m}= \frac{m+Ctv_{0}}{mv_{0}}$$

So $$(\frac{1}{v_{0}}+\frac{Ct}{m})^{-1}=\frac{mv_{0}}{m+Ctv_{0}}$$

Which integrates to;

$$\frac{m}{C}ln(m+Ctv_{0}) + C$$

However as you can see, this is apparently incorrect, and I have a feeling they're right. Just can't see where, I know I'm making a rookie mistake, but I've not practised math for months so it's quite frustrating.

Thanks in advance.

Edit:

I just thought - carrying on using my solution i could simply take $$\frac{m}{C}ln(m)$$ away from both sides, changing the constant, however that feels like such a cop-out to me, though i guess it would work. I'd still like to know how they've done it if anyone knows.

 

[fzero]

Surely;

$$\frac{1}{v_{0}}+\frac{Ct}{m}= \frac{m+Ctv_{0}}{mv_{0}}$$

So $$(\frac{1}{v_{0}}+\frac{Ct}{m})^{-1}=\frac{mv_{0}}{m+Ctv_{0}}$$

Which integrates to;

$$\frac{m}{C}ln(m+Ctv_{0}) + C$$

You've somehow identified the integration constant with C. Pick a different symbol and then evaluate the expression at t=0 to identify $$x_0$$.