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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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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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    @Jordan: $(x+1)^2=x^2+2x+1$; subtract $1$ from both sides.2012-05-31
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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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    @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...)2012-06-01