How To Integrate 1/[sqrt (x^2 + 3x + 2)] dx?

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  • #1
askor
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How do you integrate ##\frac{1}{\sqrt{x^2 + 3x + 2}} dx##?

I had tried using ##u = x^2 + 3x + 2## and trigonometry substitution but failed.

Please give me some clues and hints.

Thank you

mentor note: moved from a non-homework to here hence no template.
 

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  • #2
PeroK
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Complete the square.
 
  • #3
askor
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This is my attempt for the completing the square:

##x^2 + 3x + 2 = 0##
##x^2 + 3x = -2##
##x^2 + 3x + 2.25 = -2 + 2.25##
##x^2 + 3x + 2.25 = 0.25##
##(x + 1.5)^2 = 0.25##
##x + 1.5 = \sqrt{0.25}##
##x + 1.5 = ± 0.5##
##x_{1,2} = (± 0.5) - 1.5##
##x_1 = 0.5 - 1.5 = -1##
##x_2 = -0.5 - 1.5 = -2##

Is this correct?

What is the next step?
 
  • #4
PeroK
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I think you've solved a quadratic equation there. You were only supposed to complete the square:
$$x^2 + 3x + 2 = (x+a)^2 - b^2$$
You need to find ##a## and ##b##.
 
  • #5
askor
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I can't find the "completing the square" in my algebra book. Please tell me what page of below attached algebra book about completing the square?

https://www.sendspace.com/file/o2yqxe

The only completing the square I found in the above book is what I wrote in post #3.
 
  • #6
PeroK
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It's what i put in post #4. Google if you want.

Getting rid of the linear term might be a better description.
 
  • #7
mathman
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Try ##y=x+1.5## so you have ##\frac{dy}{\sqrt{y^2-0.25}}##. Trig substitution may work.
 
  • #8
Dr Transport
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factor the [itex] x^2 + 3x +2[/itex] into [itex] (x+2)(x+1)[/itex] and expand into partial fractions then use trig substitution.
 
  • #9
Charles Link
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factor the [itex] x^2 + 3x +2[/itex] into [itex] (x+2)(x+1)[/itex] and expand into partial fractions then use trig substitution.
I think the square root makes it so that partial fractions will not work, but trigonometric substitution after completing the square is straightforward. Edit: The trigonometric substitution is somewhat complicated. See also post 11.
 
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  • #10
mathman
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Related ##\int\frac{dx}{\sqrt{x^2-1}}=ln(x+\sqrt{x^2-1})##. I don't see any obvious simple method.
 
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  • #11
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Related ##\int\frac{dx}{\sqrt{x^2-1}}=ln(x+\sqrt{x^2-1})##. I don't see any obvious simple method.
The trigonometric substitution ## x=\sec{\theta} ## will work. I used the word straightforward in post 9, but upon working through it in detail, I see it involves computing ## \int \sec{\theta} \, d \theta ## which involves a "trick", so it is somewhat complicated.
 
  • #12
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Related ##\int\frac{dx}{\sqrt{x^2-1}}=ln(x+\sqrt{x^2-1})##. I don't see any obvious simple method.
##x = \cosh u##

##\cosh^{-1}x = \ln(x + \sqrt{x^2 - 1})##
 
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  • #14
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I can't find the "completing the square" in my algebra book. Please tell me what page of below attached algebra book about completing the square?
I looked at the link you posted, but I would have needed to download the book, and then search through it, which I don't care to do.

In your book, find the chapter or section that presents the Quadratic Formula. In that section of somewhere before it, they should talk about "Completing the Square." If your book lays out a derivation of the Quadratic Formula, it's done using the technique of completing the square.
 
  • #15
PeroK
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I looked at the link you posted, but I would have needed to download the book, and then search through it, which I don't care to do.

In your book, find the chapter or section that presents the Quadratic Formula. In that section of somewhere before it, they should talk about "Completing the Square." If your book lays out a derivation of the Quadratic Formula, it's done using the technique of completing the square.
It's all over the Internet. Here, for example:

https://en.wikipedia.org/wiki/Completing_the_square
 
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  • #16
Grasshopper
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I can't find the "completing the square" in my algebra book. Please tell me what page of below attached algebra book about completing the square?

https://www.sendspace.com/file/o2yqxe

The only completing the square I found in the above book is what I wrote in post #3.
Hi! Disclaimer: I’m a student.

To complete the square, take the quantity x + half of the second term and square it. Then add the appropriate constant that keeps the polynomial the same.

For example:

##x^2 + bx + c = (x + \frac{b}{2} )^2 + c - \frac{b^2}{4}##
 
  • #17
Grasshopper
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Hey what do you do after completing the square? Factoring this is actually really easy. Even doing a partial fraction is cake, under the radical. But I can’t see how that helps. Unless... I’m hoping completing the square leaves it in a useful form. Hmm...

I think you can get a difference of squares under the radical, since ##\frac{1}{4}## is a perfect square.

Maybe trig substitution then? Once you get it in the form of ##\frac{1}{\sqrt{a^2 - b^2}}##

But in this particular case I think a is actually a(x), so I think you’d have to do sub and then a trig sub, maybe.

This is a toughy.


Edit - actually scratch that, cause I think only the form ##\frac{1}{\sqrt{1- x^2}}## would be useful for that.

Feel free to delete if this adds nothing. Fun problem.
 
  • #18
Charles Link
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See posts 10, 11, and 12. The substitution of post 12 is perhaps the easiest.
 
  • #19
Grasshopper
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Thanks Charles. The only reason I didn’t utilize that was because it felt like it came from an integral table. But the sec substitution is probably one I should know, so disregard this haha.
 
  • #20
Charles Link
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Thanks Charles. The only reason I didn’t utilize that was because it felt like it came from an integral table. But the sec substitution is probably one I should know, so disregard this haha.
Yes, and to then do the complete calculation, you wind up integrating ## \int \sec{\theta} \, d \theta ##. That one involves a trick of multiplying numerator and denominator by ## \sec{\theta}+\tan{\theta} ## to get ## \int \frac{ d (\sec{\theta}+\tan{\theta})}{\sec{\theta}+\tan{\theta}}=\ln|\sec{\theta}+\tan{\theta} | ##. It's basically a trick, and without seeing it, you wouldn't be expected to be able to solve the integral ## \int \sec{\theta} \, d \theta ##.
 
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  • #21
askor
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I finally found how to completing the square of ##x^2 + 3x + 2##, and here is the result:

##(x + 1.5)^2 - (\sqrt{0.25})^2##

Am I correct?

If it correct, then

##\int \frac{1}{\sqrt{x^2 + 3x + 2}} dx##
## = \int \frac{1}{\sqrt{(x + 1.5)^2 - (\sqrt{0.25})^2}} dx##

Let ##u = x + 1.5##, then ##du = dx##

## = \int \frac{du}{\sqrt{u^2 - (\sqrt{0.25})^2}}##

What next?
 
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  • #23
askor
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You might try computing ## \sqrt{0.25} ##. The result is simple.

##\sqrt{0.25} = 0.5##

so

## = \int \frac{du}{\sqrt{u^2 - (\sqrt{0.25})^2}}##
## = \int \frac{du}{\sqrt{u^2 - (0.5)^2}}##

What next?
 
  • #24
Charles Link
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See posts 10,11, and 12. You also need the substitution ## u=x+1.5 ##
 
  • #25
Charles Link
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What next?
The simplest next step is essentially given by @PeroK in post 12: ## v=.5 \cosh{u} ##, but you need to show more effort here. PF rules are that the Homework Helper cannot supply the solution, and that is nearly happening here. See if you can solve it from here.
 
  • #26
vela
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Yes, and to then do the complete calculation, you wind up integrating ## \int \sec{\theta} \, d \theta ##. That one involves a trick of multiplying numerator and denominator by ## \sec{\theta}+\tan{\theta} ## to get ## \int \frac{ d (\sec{\theta}+\tan{\theta})}{\sec{\theta}+\tan{\theta}}=\ln|\sec{\theta}+\tan{\theta} | ##. It's basically a trick, and without seeing it, you wouldn't be expected to be able to solve the integral ## \int \sec{\theta} \, d \theta ##.
There's a less tricky way to integrate that.
$$\int \sec \theta\,d\theta = \int \frac 1{\cos \theta}\,d\theta = \int \frac {\cos\theta}{\cos^2 \theta}\,d\theta = \int \frac {\cos\theta}{1-\sin^2 \theta}\,d\theta$$ then use ##u=\sin\theta##.
 
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  • #27
askor
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The simplest next step is essentially given by @PeroK in post 12: ## v=.5 \cosh{u} ##, but you need to show more effort here. PF rules are that the Homework Helper cannot supply the solution, and that is nearly happening here. See if you can solve it from here.

Why use hyperbolic cos instead of standard/regular cos?
 
  • #28
askor
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There's a less tricky way to integrate that.
$$\int \sec \theta\,d\theta = \int \frac 1{\cos \theta}\,d\theta = \int \frac {\cos\theta}{\cos^2 \theta}\,d\theta = \int \frac {\cos\theta}{1-\sin^2 \theta}\,d\theta$$ then use ##u=\sin\theta##.

You said use ##u = \sin \theta##, then the form will be changing like this:

##\int \frac{\cos \theta}{1 - u^2} d\theta##

What should I do with the ##\cos \theta##?
 
  • #29
vela
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Why use hyperbolic cos instead of standard/regular cos?
Because you can. Try it both ways and see why you might choose one over the other.

What should I do with the ##\cos \theta##?
Ask again after you finish rewriting everything in terms of ##u##.
 
  • #30
askor
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Ask again after you finish rewriting everything in terms of ##u##.

I see.

If ##u = \sin \theta## then ##du = \cos \theta \, d \theta##

Then the form will be like this:

##\int \frac{du}{1 - u^2}##

What should I do next?

My brain is fatigue.
 
  • #31
vela
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Take a break and try again later with fresh eyes.
 
  • #32
PeroK
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There's a less tricky way to integrate that.
$$\int \sec \theta\,d\theta = \int \frac 1{\cos \theta}\,d\theta = \int \frac {\cos\theta}{\cos^2 \theta}\,d\theta = \int \frac {\cos\theta}{1-\sin^2 \theta}\,d\theta$$ then use ##u=\sin\theta##.
Using ##\cosh## is about a hundred million time better. If you'll excuse the hyperbole!
 
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  • #33
askor
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Using ##\cosh## is about a hundred million time better. If you'll excuse the hyperbole!

How do you use ##\cosh##? Please show me an example.
 
  • #34
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How do you use ##\cosh##? Please show me an example.
Have the hyperbolic trig functions been presented to you, yet? If not, an example probably won't make much sense.
 
  • #35
Vanadium 50
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If you'll excuse the hyperbole!

I saw what you did there.
 

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