<s>[INST] Evaluate $\int_0^{\sqrt{5}} \sqrt{4x^4 + 16 x^2} dx$ $$\int_0^{\sqrt{5}} \sqrt{4x^4 + 16 x^2} dx$$
The square root really got me confused here. I've tried using the standard trick with $x = x + \sqrt{x^2+4}$ but failed. That might still be the best approach.
What's the best way to solve this and how to do it?
 [/INST] For real $x,$
$$\sqrt{4x^4+16x^2}=\sqrt{4x^2(x^2+4)}=2|x|\sqrt{x^2+4}$$
As $x\ge0,$ $$\sqrt{4x^4+16x^2}=2x\sqrt{x^2+4}$$
Set $x^2+4=u$ or $x^2+4=u^2$
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