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September 12th, 2017, 06:52 PM   #1
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Question 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
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September 12th, 2017, 07:30 PM   #2
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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.
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September 13th, 2017, 05:01 AM   #3
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"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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