Is this one sentence right?

In my opinion, the first one is the correct idiomatic choice, but the second one, with the infinitive particle, is not necessarily incorrect.

"Let's go study history." Ok

"To" is not necessary. In fact, it is wrong. You may say "Let's go to the park". But when we use "let's go" which is followed by another verb, i.e. "swim", "see" (a movie), or "take" (a walk). leave the prep "to" out.

Marold"Let's go study history."or"Let's go to study history."

Leaving out to after go is very common in informal English even though traditional grammarians may frown upon it. Let's has nothing to do with the absence or presence of to here. Let's go and study history is another correct option. However, I

go study belongs to the "go do" family of expressions.

Read more about it here:

CJ

Maybe CJ would be kind enough to shed some light on this......