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I just wonder: How do you know whether an integral can be solved? For example, exponential integral can not be derived to final result.

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What does final result mean? – Emanuele Paolini Feb 12 at 9:02

marked as duplicate by user7530, Ittay Weiss, Gerry Myerson, Hans Lundmark, Michael Greinecker Feb 12 at 9:35

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You don't solve integrals, you compute them. The integral $$ \int_0^x \frac{e^t}{t}\, dt $$ can be computed as any other integral. Only the result is not a function which you may find in every pocket calculator...

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indefinte integrals are not computed but evaluated. – Arjang Feb 12 at 9:20
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Integrals are not solved, indefinite integrals are evaluated. One may compute definite integrals. The question you are trying to ask is this: "What integrals can be evaluated in terms of elementary or well known transcedental functions"

Example of transcedental functions : $\sin x , \ln x, e^x $ etc.

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$Ei(x)$ is a well known transcendental function. – Emanuele Paolini Feb 12 at 9:39
The distinction definite/indefinite integrals regarding evaluation/computation seems irrelevant since lots of indefinite integrals are equivalent to definite ones and vice versa. – Did Feb 12 at 9:40
@Did : yes that is correct, but I have not seen any examples of the two being interchanged in books, if I see some examples of them being interchanged in books written by native English speakers then I can agree. – Arjang Feb 12 at 10:11
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Your procedures of agreement are a tad peculiar, to say the least... Since you do not mention books written by native English speakers in your post, my remark fully applies and you might want to reformulate your (at present misleading) post. – Did Feb 12 at 10:15
@Did : If you could adjust the misleading part any way you see fit please, Ill appreciate it. I agree with anything that I either can infer, deduce or see some examples of it around. PS : I did found this answer of yours awesome : math.stackexchange.com/questions/74347/… – Arjang Feb 12 at 10:24
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There is no universal criterion, distinguishing which integrals can be computed in elementary functions. (Rational functions of the exponent.)

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