My Math Forum Integration by recognition

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 September 12th, 2017, 06:52 PM #1 Newbie   Joined: Sep 2017 From: Australia Posts: 2 Thanks: 0 Integration by recognition Hi, I've been stuck on integration by recognition questions for a long time and can't get my head around them. Any help would be greatly appreciated! Thanks in advance. Here's a couple of examples: 1. Differentiate ln(3-2x) and hence find the antiderivative of 5/(3-2x) 2. Show that (3-2x)/(1-x)=2+1/(1-x) and hence find the antiderivative of (3-2x)/(1-x) Last edited by skipjack; September 13th, 2017 at 12:48 AM. Reason: Added brackets where appropriate
 September 12th, 2017, 07:30 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 1,696 Thanks: 861 I'm not sure what you mean by integration by recognition, but let's take a look. $\dfrac{d}{dx} \ln(3-2x) = \dfrac{-2}{3-2x}$ $\dfrac{5}{3-2x} = \left(-\dfrac{5}{2}\right) \dfrac{-2}{3-2x}$ so $\displaystyle \int ~\dfrac{5}{3-2x}~dx = \left(-\dfrac{5}{2}\right)\int~\dfrac{-2}{3-2x}~dx = \left(-\dfrac{5}{2}\right) \ln(3-2x)$ Does this make sense? See whether you can do the 2nd one. Last edited by skipjack; September 13th, 2017 at 12:49 AM.
 September 13th, 2017, 05:01 AM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 2,922 Thanks: 785 "Integration by recognition" means to integrate by "recognizing" the integrand as being the derivative of another function. Of course, that depends upon what derivatives you know! Here, they asked you to find the derivative first so you would have to know it.

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