Idea to show $\max\left\{\int_{0}^{1}\sqrt{1+n^2x^{2n-2}} \right\}dx < 2$

Question

Let $n \in \mathbb{N}$ ,$$I_n=\int_{0}^{1}\sqrt{1+n^2x^{2n-2}}dx$$ we want to prove $$\forall n\in\mathbb{N} :I_n\leq2$$ I am looking for new Idea(s) or alternative proof.Thanks in advance.

Answer 1

Take $$y=c_n=x^n\\x\in[0,1] ,n\in\mathbb{N}$$ all of the $c_s$'s pass two points(0,0),(1,1),for example for n=1,2,3,4 as below . $$x\in[0,1],\forall n\in \mathbb{N} : c''_n\geq 0 \\and , x^{n+1}<x^{n}$$ so $$length_{c_1}<length_{c_2}<length_{c_3}<...<length_{c_n}<OH+HM=2$$ length of curve formula was $$\int_{0}^{1}\sqrt{1+(y')^2}dx$$ in this case we have $$\forall n \in \mathbb{N} :length_{c_n} < 2\\so\\I_n=\int_{0}^{1}\sqrt{1+(y')^2}dx=\int_{0}^{1}\sqrt{1+(nx^{n-1})^2}dx<2\\I_n<2$$

Answer 2

For every $a, b \ge 0$, we have $\sqrt{a+b} \le \sqrt{a} + \sqrt{b}$. Then $I_n \le 1+\int_0^1 nx^{n-1}dx = 2$