you can make this work (i would have probably tried a trig substitution, but your way is fine, probably easier).

note that this is the same as: you'll see why i did that in a minute.

Let

So, our integral becomes

by partial fractions we find that this is:

and i think you can take it from there, right?

(by the way, there's actually a formula for this integral, look it up in your text book. it will be an equivalent answer. i think it will be arcsec(x) or something like that)