# Arc Length/Help with Integral

#### miglo

1. The problem statement, all variables and given/known data
$$y=\frac{1}{3}\left(x^2+2\right)^{3/2}$$

2. Relevant equations
$$\int_{a}^{b}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}dx$$

3. The attempt at a solution
$$\frac{dy}{dx}=x\sqrt{x^2+2}$$
$$\int_{0}^{3}\sqrt{1+\left(x\sqrt{x^2+2}\right)^{2}}dx=\int_{0}^{3}\sqrt{1+x^4+2x^2}dx$$
I'm stuck with that integral, not sure what the antiderivative for that function would be
Any hints at the next step would be greatly appreciated.

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#### Char. Limit

Gold Member
Here's a hint: x4+2x2+1 = (x2+1)2

#### miglo

Wow I can't believe i didn't see that.
I tried Wolfram Alpha earlier and they used a substitution but i couldn't follow their steps so i decided to consult the Physics Forums.
Thanks Char. Limit.

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