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I am trying to find the integral $$\int \sqrt{x^2 + 2x}$$

I know that u substitution won't work and I don't see integration by parts working either, I can't make a trig subsitution because the radicand isn't in the form of $a^2 -x^2$ $a^2 +x^2$ or $x^2 - a^2$

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2 Answers

up vote 3 down vote accepted

Complete the square: $x^2+2x=(x+1)^2-1$. Now let $u=x+1$, and ...

Added: If we set $u=x+1$, then $du=dx$, and

$$\int\sqrt{x^2+x}\,dx=\int\sqrt{(x+1)^2-1}\,dx=\int\sqrt{u^2-1}\,du\;.$$

In your question you mentioned the kind of substitution that handles this kind of integral.

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I don't understand, the square root of $(x+1)^2$ is x+1 so my answer is $x^2 /2$? –  user138246 May 31 '12 at 11:14
    
@Jordan: Don't forget the $-1$. –  mixedmath May 31 '12 at 11:18
    
What is the -1 from? –  user138246 May 31 '12 at 11:19
    
@Jordan: $(x+1)^2=x^2+2x+1$; subtract $1$ from both sides. –  Brian M. Scott May 31 '12 at 11:20
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HINTS

  1. Complete the square
  2. Do u-substitution (but pay attention to what your $u$ is) to turn it into a form you know
  3. Trig Sub
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I have forgotten about complete the square, would that be the only way? –  user138246 May 31 '12 at 11:13
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@Jordan: Is it the only way? Likely not. But it's the most standard way, and the way I would students to do it. And it's the only way that jumps out at me. –  mixedmath May 31 '12 at 11:14
    
It just seems like a pretty complicated procedure to learn to do a problem like this when I already have no much else to learn. –  user138246 May 31 '12 at 11:15
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@Jordan: I think that they expect completing the square to be in your toolbox of skills already. It will certainly come up again and again in your calculus studies, as it's a really standard technique. Perhaps now would be a good time to internalize it. –  mixedmath May 31 '12 at 11:18
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@Jordan: You need to do something about your studing technique. Have you thought about trying spaced repetition software (like Anki) to help you memorize things? It's really helpful! (And by the way, there's no excuse for not remembering such a simple thing as how to complete the square. It ought to take you less time to memorize it once and for all than what you have already wasted on this single integral by not remembering how to do it...) –  Hans Lundmark Jun 1 '12 at 7:07
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