"bit of investigation" is a correct and usable phrase in written English.
You can use it to describe a small amount of research or inquiry into something. For example: "I did a bit of investigation into the matter, but I couldn't find any conclusive answers."
With a bit of investigation I realised my lighting was drawing a lot of electricity.
A bit of investigation disclosed that a nephew had asked a firm of boatbuilders to send a catalogue to "my Uncle Fred", as a broad hint.
Having done a bit of investigation after I had finished it I have discovered that it was written especially with dyslexic or reluctant readers in mind.
After a bit of investigation, he found that his credit scores had fallen 125 to 150 points, depending on which bureau he asked.
After a bit of investigation I found a two-week course in the development of French cuisine at l'Institut Parisien de Langue et de Civilisation Françaises, in the 15th Arrondissement, that seemed to offer what I was looking for.
But whatever the reason for the modesty, this neighborhood of overflowing plates and Latin rhythms opens up with a little bit of investigation, and some of its secrets are surprising: a jazz club in a living room, for example, and a debutante's vandalism.
I had to do a bit of Internet investigation to find out exactly what I'd done.
The Indonesian capital is notorious for hectic activity and gridlocked streets, but a bit of further investigation reveals there's plenty more to Jakarta than traffic jams.
Don't trust a likeness without a bit of further investigation.
IA: Well, I had to do quite a bit of personal investigation with the inter-figures of... Brick 1 to see how far I wanted to make any cross-references between the two.
Eventually I became old enough to do a bit of reading and investigation.
