# Integral of $\int \sqrt{x^2 + 2x}$

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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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
2. Do u-substitution (but pay attention to what your $u$ is) to turn it into a form you know