Question Number 1472 by 123456 last updated on 11/Aug/15
![find f:[0,1]→R such that a.∫_0 ^1 fdt=∫_0 ^1 (√(1+((df/dt))^2 ))dt b.∀x∈[0,1] ∫_0 ^x fdt=∫_0 ^x (√(1+((df/dt))^2 ))dt](https://www.tinkutara.com/question/Q1472.png)
$$\mathrm{find}\:{f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R}\:\mathrm{such}\:\mathrm{that} \\ $$$${a}.\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{fdt}=\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\sqrt{\mathrm{1}+\left(\frac{{df}}{{dt}}\right)^{\mathrm{2}} }{dt} \\ $$$${b}.\forall{x}\in\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\underset{\mathrm{0}} {\overset{{x}} {\int}}{fdt}=\underset{\mathrm{0}} {\overset{{x}} {\int}}\sqrt{\mathrm{1}+\left(\frac{{df}}{{dt}}\right)^{\mathrm{2}} }{dt} \\ $$$$ \\ $$
Commented by prakash jain last updated on 13/Aug/15
$${f}\left({t}\right)=\mathrm{cosh}\:{t} \\ $$$${f}\left({t}\right)=\mathrm{1} \\ $$