I think this question is not research-level, but nevertheless, since the answer is not generally known in the mathematical community, here it is.

You are mistaken. There is a "formula" for integration. It goes under the name "Risch algorithm". That is, there is a completely mechanical procedure which computes the closed form of an indefinite integral of a function which is a composition of elementary functions, if there is one and fails otherwise. However, the procedure is quite a bit more complicated, and is thus not generally taught in schools. In fact, it is so complicated that complete implementations are hard to come by. Caveat: there are parts of Risch's algorithm which are not known to be entirely algorithmic (see Wikipedia for a discussion of this point).

The general theory and problem of which the Risch algorithm is the solution was developed already by Liouville.

There is also a more simplistic answer. Many rules of differentiation can be inverted to give corresponding rules of integration, for example:

- The product rule becomes integration by parts.
- The chain rule becomes the substitution rule.
- Integral of $x^n$ is deduced from the derivative of $x^{n+1}$.
- Integral of $\sin x$ is deduced from the derivative of $\cos x$, etc.

The trouble is, while the rules of differentiation provide a complete set of rules for getting rid of all the derivatives in a systematic fashion, the corresponding rules for integration do not. So many think that integration is an art. It isn't, it's an algorithm invented by Risch.