Looking on the internet deeply has found these results:

I will do better. is the most popular phrase on the web.

I will do the best I can 1636,000 results on the web	I will do better. 1728,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
I will do the best I can in the time that we have left. Darryl, believe me, I will do the very best I can. I do the best I can, Detective. I will do the best I can, but I am not going to lie to you. I do the best I can. Look, I will do the best that I can here, which I realize isn't much, but you have to choose between us and Charlie, and I am asking you to choose Charlie. I will compete and I will do the best I can. I do the best I can, Mistress.	I am so sorry I can't look at her and say... that I will promise to do better. I will never do better than you. I think it will do better abroad. I can and will do better. Mr Bangemann has already promised this House that he will do better. I can and will do better. I promise I will try to do better.

I will do the best I can

1636,000 results on the web

I will do better.

1728,000 results on the web

More popular!

I will do the best I can in the time that we have left.
Darryl, believe me, I will do the very best I can.
I do the best I can, Detective.
I will do the best I can, but I am not going to lie to you.
I do the best I can.
Look, I will do the best that I can here, which I realize isn't much, but you have to choose between us and Charlie, and I am asking you to choose Charlie.
I will compete and I will do the best I can.
I do the best I can, Mistress.

I am so sorry I can't look at her and say... that I will promise to do better.
I will never do better than you.
I think it will do better abroad.
I can and will do better.
Mr Bangemann has already promised this House that he will do better.
I can and will do better.
I promise I will try to do better.