What is the arc length of $f(x)= xsqrt(x^3-x+2)$ on $x in [1,2]$ ?

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Answer 1 · 2018-07-12T16:16:45

$approx 4.3712$

We are using the formual
$\int_a^bsqrt(1+f'(x)^2)dx$

by the product and the chain rule we get

$f'(x)=sqrt(x^3-x+2)+x*1/2*(x^3-x+2)^(-1/2)*(3x^2-1)$

this is

$f'(x)=(2*(x^3-x+2)+3x^3-x)/(2sqrt(x^3-x+2))$

simplifying we get

$f'(x)=(5x^3-3x+4)/(2sqrt(x^3-x+2))$
and we have to integrate

$int_1^2sqrt(1+((5x^3-3x+4)/(2sqrt(x^3-x+2)))^2)dx$
By a numerical method we get $4.3712$

What is the arc length of f(x)= xsqrt(x^3-x+2) f(x)= xsqrt(x^3-x+2) on x in [1,2] x in [1,2] ?