Friday, January 6, 2017

Arc length of the curve y=ln(x)



Q: Find the arc length of the curve y=ln(x) where x ranges from 3 to 15.



I think I am stuck in calculation part.



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The answer is 2+ln(3)12ln(5). But I can't derive that from my last line.



help me, please.


Answer



Your problem now is how to evaluate the integral 153x2+1xdx.



Let
F(x)=x2+1xdx.
Let x=tanθ. All your computations after this substitution are all correct.

Note that cscθ=x2+1x cotθ=1x and
secθ=x2+1.
Hence, we get
F(x)=ln(x2+1+1x)+x2+1+C.
Thus,
153x2+1xdx=F(15)F(3)=[ln(515)+4+C][ln(33)+2+C]=ln(33)ln(515)+2=ln[33÷515]+2=ln[33155]+2=ln[3155]+2=ln[3115]+2=ln(35)+2=ln3ln5+2=ln312(ln5)+2.


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