Hi Suvadip !
Let consider a simple example :
Question : integrate dx/x ?
Answer : ln(x) +constant. But, this supposes that the function ln(x) is known. If one doesn't know ln(x) he cannot answer.
Then, consider a more difficult example :
Question : integrate dx/sqrt(sin(x)) ?
The answer involves an "elliptic" function. But, this supposes that the elliptic functions are known. If one doesn't know this kind of functions, he cannot answer.
All the integrals cannot be expressed in terms of the elementary functions.
Some functions, which are less usual as the elementary functions, are called "special functions". They are a lot, some of rather low level (such as the elliptics), some of much higher level (such as hypergeometrics), and many are even not referenced.
A paper for the general public : "Safari in the Contry of Special Functions" (pp.18-36) translated from "Safari au pays des fonctions spéciales" :
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