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So I'm currently trying to solve

$$\int \sqrt{ 1+\frac{1}{3x} } \, \, dx$$

I know that this can also be represented as ((x+1/3)/x)^1/2 but I dont like that form. I also know that this can be done with sustitution. I've done lot's of stuff but I get stuck everytime. You don't need to solve me the problem, just point me to the right direction if you want to and I'll finish it myself.

I'll write my first impression. I proceed to choose $u = x + \frac{1}{3x^{2}}$

so $du =1 -1/3x^{2} dx$

So I end with $\int \sqrt{u} \, \, \, du -3x^{2} $

I'm sure this is wrong but I don't know why. Maybe it wasn't wise to choose u as the whole square root? I did so because the immediate integral of x^1/2 is easy. Did I do something wrong? Or can I continue from here? If so, how?

Thanks a ton!!! =)

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if you use that u, make sure you get the power right: $du = 1-\frac{1}{3}x^{-2} dx$ –  Alex Oct 13 '12 at 18:32
Both the OP and @Alex write the right hand side of $du$ without parentheses, and it is wrong in my opinion. –  enzotib Oct 14 '12 at 12:35
@enzoib, yes it is, sorry. can't edit anymore though –  Alex Oct 14 '12 at 12:39
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4 Answers 4

up vote 2 down vote accepted

$$\int \sqrt{ 1+\frac{1}{3x} } \, \, dx$$ let $u^2=3x \implies 2u\ du = 3\ dx$ $$\frac{2}{3}\int \sqrt{ 1+\frac{1}{u^2} } \, \, u\ du$$ $$\frac{2}{3}\int \sqrt{ u^2+1 } \, \, du$$ now you can use $u = \sinh(t)$

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This works nicely, unless we are worried about negative $x$, in which case we need to do a cases analysis. –  André Nicolas Oct 13 '12 at 18:56
you're my effing hero –  Damieh Oct 13 '12 at 18:57
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Well, the problem appears to be the radical. I'm not sure setting $u$ equal to everything under the radical looks promising, as you still need a $du$. There is, however, more than one way to get rid of a radical.

If you are comfortable with hyperbolic functions, letting $\frac1{3x}=\sinh^2 u,x=\frac13\operatorname{csch}^2u$. Myself, I was never taught those and would go with the similar trig substitution $\frac1{3x}=\tan^2u,x=\frac13\cot^2u,dx=-\frac23\cot u\csc^2udu$.

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As suggested by busman, it may be a good idea to let $u=3x$. The constants one has to drag around are then less annoying. So, apart from a constant factor, we want $$\int\sqrt{1+\frac{1}{u}}\,du.$$ Now I would suggest letting $w^2=1+\dfrac{1}{u}$. Then $$2w\,dw=-\frac{du}{u^2}=-(w^2-1)^2 \,du.$$ We end up having to find something like $$\int-\frac{2w^2\,dw}{(w^2-1)^2}.$$ This is a not completely pleasant partial fractions problem.

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WOW man, that's awesome!! That's waaay over my level though =P –  Damieh Oct 13 '12 at 18:43
One can do better. As suggested by busman in a comment, one can alternately integrate by parts, one part $-2w$, the other part $\frac{w}{(w^2-1)^2}$. Then we end up basically wanting $\int \frac{w^2\,dw}{w^2-1}$, which by division basically leaves us with $\int\frac{dw}{w^2-1}$, easy partial fractions. Instead of my $w^2$, we could make the hyperbolic substitution $\sinh^2 y=1+\frac{1}{u}$, works nicely but too magical. –  André Nicolas Oct 13 '12 at 18:52
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I suggest you to set $u = 3x$, but prior to that, do some thing with the sum under the root. Then use a pretty elementary integration technique.

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So I did, and I ended with 1/3 integrate ((u+1)/u)^1/2 du.... and I'm stuck again =( –  Damieh Oct 13 '12 at 18:36
Yep, now split this expression in some imaginative way... –  busman Oct 13 '12 at 18:37
okay I'm trying, I'll let you know. Thanks a lot!! =) –  Damieh Oct 13 '12 at 18:39
Look at André Nicolás' idea too, it seems to me that using parts integration after his final statement would work. –  busman Oct 13 '12 at 18:40
I forgot to refresh the page lol =P. It's way over my level though. Anyway the only way I can split the expression is by distributing the roots. I can't find another way (I'll keep thinking though). –  Damieh Oct 13 '12 at 18:44
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