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I need to find the anti-derivative of $f(x)=g(x)\cdot{}h(x)$. How do I do that?

3 Answers3

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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 $\frac{\sin x }{x}$.

davidlowryduda
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For the product of some functions, you can use integration by parts. The link should help give you some insights as to when that's appropriate.

In your case, there's no way of knowing the relation between arbitrary functions, but there often is a relation between the product of two functions, as you'll see, when integration by parts is the tool of choice!

amWhy
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  • Wowwww. A nice Avatar Amy. – Mikasa Mar 26 '13 at 15:06
  • I need a help, can you tell me how to put the link in the answer so that it appears as you did here with " integration by parts" and not as an URL? –  Mar 26 '13 at 15:24
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    Hi, @Thus: you can manually put name you want to give in brackets, then immediately follow with the URL in parentheses [integration by parts](http://tutorial.math.lamar.edu/Classes/CalcII/IntegrationByParts.aspx) as in the case above, or simply click on the image icon in at the top left of your answer, which will prompt you to enter a URL, and then highlights the area (bracketed) to enter the link's reference. – amWhy Mar 26 '13 at 15:31
  • Than you! Thank you very much! :-D –  Mar 26 '13 at 15:36
  • You're welcome, @Thus ! – amWhy Mar 26 '13 at 15:37
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There is no general formula for the anti-derivative of arbitrary functions. If you have specific functions in mind, try integration by parts.

gerw
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