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When doing differentiation, I know that if $f$ is a function on $x$, then

$ { d \over dx } f^2 = 2 f {df \over dx} $

so the opposite in integration is also clear:

$ \int 2 f { df \over dx } dx = f^2 $

I also know that

$ \int x^2 dx = { x^3 \over 3} $

But I'm not sure as to how I can evaluate:

$ \int f^2 dx $

I mean is there any identity for this? That the above is equal to another function of $f$ (such as $f^3 \over 3$ times something)? Is there any method to find this? I googled some but perhaps I wasn't using proper search terms so I didn't get any clear results so I'm asking here. [I hope my question is clear enough :-(]

2 Answers 2

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Can't be done, in general. For example, it is easy to do $\int xe^{x^2}\,dx$ but there is no expression for $\int x^2e^{2x^2}\,dx$ in terms of the familiar functions of undergraduate mathematics.

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    Yes --- and, no. In higher level studies you just define a new function, the "error function", by ${\rm erf}(x)=\int_{-\infty}^xe^{-t^2}\,dt$ (or something like that), and then you can express the second integral in terms of erf.2012-11-26
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i suppose if you substitute the function as any arbitrary variable X u can solve this integration. for example while integrating (1+x)^2 you substitute 1+x=t then differentiate both with respect to t. this gives you dx=dt. here, no matter what your function, make dx the subject of the equation so that you can replace dx in your original integral. after that you integrate the function as integration of t^2 which is t^3/3 and then resubstitute t=1+x in the answer to get your answer.This is of course a very simple example but the process is the same.

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    it works in most cases where the function isn't really complicated. i really think the person with the query should at least try this....2014-05-25