Utility Function, which one I use?

In many of economics books that relate to the DSGE model use different Utility Function like

and there are others

so Which one should I use? and why

That depends on what you are trying to model. The function you wrote down implies constant relative risk aversion and an intertemporal elasticity of substitution that it the inverse of the coefficient of relative risk aversion. This gives you some desirable and some undesirable features. Whether you can live with those properties depends again on what you are trying to achieve. It will for example not be useful if you are thinking about asset pricing.

Thank you so much for your reply
I need an example for using the tow functions, to understand that, I wrote many researches using them, but they didn’t point out why they choose it

Which two functions? See also https://economics.stackexchange.com/questions/1813/why-is-crra-utility-often-used-in-macroeconomics-dsge-model

I read it before and understood it, I will change the question

when we should use the first equation, and when we should use the other

Regarding your first two equations: the first one is a felicity or period utility function. The second one is a (lifetime) utility function. Think of the integral as a sum. It is the sum of felicity in each period, appropriately discounted.

I know that, but you gave me an explanation about the two functions. the question in another way, for an example, what should be the search title which adopt the first utility function, and the other one title which adopt the second function.

Sorry, but I do not understand your question.

for instance, if you want to conduct research what is the search title you will choose if you want to use the first utility function, and what is the other research title if you choose the other utility function, because there are many rules to choose the first utility function rather than the second utility function in DSGE Model

You are still not getting my point. One is the period utility function, the other one the lifetime utility function. They capture different aspects, i.e. the intertemporal and the static within period perspective. DSGE models are by nature dynamic, so you will always have the lifetime utility function appearing.

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