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 $x^2 – a^2$
Best Answer
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.