How to know if a function is non-integrable without trying to solve it?

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so for example, this problem wanted to know if a differential equation can be solved using laplace transform. at the end there was an integral, but in the solution they simply conclude that this integral cannot be found. How exactly does one know that? In an exam i won't be having all the time in the world, so i can't just keep trying to solve it enter image description here enter image description here