why isnt there answer to integral of the form $\int \frac{u}{v}$

Question

Why is there no general form for the $$\int \frac{u}{v}$$

The idea why I thought about this is becausewe can differentiate a function of the form $u/v$ means its some other functions integral so there might be a remote probability that there is some way to get the integral of the form $u/v$.

Or might someone prove that there can’t be an integration done by some general form. Thanks. Guide me to other question if there exists such question.

Answer 1

Your problem is similar to integrating $f(x)g(x)$. So here, I will leave you a link to this question and the accepted answer.

There is no generally easy way. For example, we know the antiderivative of both $\sin x$ and $\frac{1}{x}$, but there is no elementary antiderivative of their product $\frac{\sin x }{x}$.

Of course, as the second answer in the link points out, you could use integration by parts.

If $w=\frac{1}{v}$, then $$\int \frac{u}{v}=\int uw$$

From where you can proceed.