Why isn't there any general form for the $$\int \frac{u}{v}$$

The idea why I thought about this is becausewe can differentiate a function of tge 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 cant be an integration done by some general form. Thanks . Guide me to other question if there exists such question.

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    $\begingroup$ Was there a general form for $\int uv $? If there isn't, how can there be one for $\int \frac{u}{v}$? And if there is, isn't it easy to get $\int \frac{u}{v}$? $\endgroup$ – S.C.B. Mar 27 '16 at 5:36
  • $\begingroup$ u,v can be polynomials, trigonometric stuff, anything? $\endgroup$ – Nikunj Mar 27 '16 at 5:36
  • $\begingroup$ Wolframalpha says that there is no result found in terms of standard mathematical functions... $\endgroup$ – S.C.B. Mar 27 '16 at 5:40
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    $\begingroup$ @MXYMXY were you seriously expecting a general result? $\endgroup$ – Nikunj Mar 27 '16 at 5:42
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    $\begingroup$ but $\int \frac{u}{v}$ can be worked out for most polynomials if you are willing to $\endgroup$ – Nikunj Mar 27 '16 at 5:45

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.


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