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October 28th, 2010, 03:17 PM   #1
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Impossible to integrate?

I am taking Calculus II right now, and I have an integral in my homework that just seems to be impossible to do with the techniques my class has been taught, unless there is a typo somewhere. I have to integrate (2x-3)/((4x-x^2)^1/2). Now, I have successfully integrated this function using integration by parts, as well as trigonometric substitution. However, I am technically not supposed to know how to use those methods. Basically, the only methods I am allowed to use to solve this function are normal substitution and merely knowing a function's antiderivative. So, anyone know how to solve this question with these restrictions?
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October 28th, 2010, 03:43 PM   #2
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Re: Impossible to integrate?

Let u = 4x -x^2
then -du = (2x -4)dx

Rewrite your numerator as
2x -4+1=
(2x-4)dx + 1dx

Then you have two integrals...
-du/u^(1/2) + 1/something...

Not quite sure about the second one. Probably should focus on not getting in a wreck...
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October 28th, 2010, 03:50 PM   #3
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Re: Impossible to integrate?

That's actually what I did. The first part was easy to solve. However, the second part requires trigonometric substitution. =/
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October 28th, 2010, 03:55 PM   #4
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Re: Impossible to integrate?

Then our only hope is skipjack, or the ubiquitous counter-example
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October 28th, 2010, 04:50 PM   #5
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The integral is (Merely knowing an antiderivative is allowed.)
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October 28th, 2010, 05:10 PM   #6
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Re: Impossible to integrate?

I would wager a great deal that your knowledge of antiderivatives is a wee bit more extensive than a Calc II student who's never used trigonometric substitutions.
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October 28th, 2010, 08:55 PM   #7
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How so? Most tables of standard integrals include and it's easy to see that It suffices that the original integrand can be rearranged as two terms that are easily recognized as standard forms.
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October 29th, 2010, 07:26 AM   #8
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Re: Impossible to integrate?

Ah, thanks Slipjack. I actually seem to recall something of that sort from AP Calculus. I am a bit rusty though since I took it two years ago. ;-; I think I remember two others as well, but could you verify them for me? Something about the integral of 1/(((a^2)+(x^2))^(1/2))=arcsec(x/a) and the integral of 1/(((x^2)-(a^2))^(1/2))=arctan(x/a)? I think at least, cannot remember too well. =/
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October 29th, 2010, 04:00 PM   #9
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Re: Impossible to integrate?

According to the table of integrals in my old calculus textbook:





whereas:



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