The previous answer has valuable information for you, but I would like to emphasize another aspect of the use of counter-argument in a persuasive or argumentative essay. The use of counter-argument should always be part of your strategy of persuasion because it offers you an opportunity to establish your intellectual honesty, enhance your credibility, and convey your sense of fair play. If you simply present your side and facts or ideas favorable to your point of view, those in your audience who might be aware of arguments in opposition are thinking about those counter-arguments. They are thinking, "But what about this, and what about that?" They will perceive you to be intellectually dishonest if you cannot bring yourself to admit there are other points of view, judge you as less than credible, and feel that you do not play fair. What you need to remember in a persuasive or argumentative paper is that you are not always "preaching to the converted." You are trying to persuade people who do not necessarily agree with you or have simply not made up their minds yet. But if you introduce counter-arguments, if only to knock them down, which we call the "straw man" strategy, then your audience is persuaded that you are strong enough in your conviction and support to be generous enough to talk about the other side. Do you see how powerful and persuasive that can be?
If you wish to use these phrases in an argumentative, or persuasive essay, you will employ these in your counterarguments. for persuasion must provide reasons why some ideas are logical while others are illogical. That is, persuasive writing is writing that tries to influence a reader to accept your claim; therefore, you must use strong, relevant evidence to support this claim, while refuting any counterarguments. So, you will use "while some maintain that......, and others claim that.....", history has shown that/studies have proven that/statistics on ....have proven that......etc. (Find the evidence you need to dispute opposing claims.)
Examples of this phrasing are in the speeches of Colonial Americans such as Patrick Henry and Thomas Paine. In his "Crisis No. 1," for instance, Paine mentions what some feel about the British present in the colonies:
There are persons too who see not the full extent of the evil which threatens them; they solace themselves with hopes that the enemy, if he succeeds, will be merciful. [ This is a "some claim" statement]
Later, Paine counters the claims by asking a rhetorical question; his is a question so logical that it requires no answer:
Is this the madness of folly, to expect mercy from those who have refused to do justice; and even mercy, where conquesst is the object, is only a trick of war....
See the how-to topics on enotes for more help. One is cited below that will greatly assist you.