What is the arclength of $f (x) = x^{2} e^{\frac{1}{x}}$ $f(x)=x^2e^(1/x)$ on $x \in [1, 2]$ $x in [1,2]$ ?

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Answer 1 · 2018-07-14T19:56:35

$\approx 16.4316$ $approx 16.4316$

We are using the Formula

$int_a^bsqrt(1+(f'(x))^2)dx$
Calculating $f'(x)$

$f'(x)=2xe^(1/x)+x^2*e^(1/x)(-1/x^2)$
simplifying we get

$f'(x)=e^(1/x)(2x-1)$
so we have to compute

$int_1^2sqrt(1+(e^(1/x)(2x-1))^2)dx$

This integral can not expressed by the known elementary functions.

By a numerical method we get $approx 16.4316$

What is the arclength of f(x)=x^2e^(1/x)f(x)=x2e1xf(x)=x^2e^(1/x) on x in [1,2]x∈[1,2]x in [1,2]?