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#