Finding a definite integral from the Riemann sum

In summary, the conversation discusses an interval with given values, and using those values, the expression (1 + i/n)(1/n)ln[(n + i)/n] is rearranged using logarithmic properties. The question of what goes in the logarithmic function is raised, and it is suggested to simplify the argument of the logarithm. Eventually, it is realized that the argument can be simplified to (1 + i/n), resulting in the function f(x) = xlnx + (1/n)ln(1+i/n).
  • #1
crememars
15
2
Homework Statement
Consider the following limit of a Riemann sum for a function f on [a, b]. Identify f, a, and b,
and express the limit as a definite integral.

*see actual expression in the description below. it was too complicated to type out so I included a picture instead.
Relevant Equations
∆x = (b-a)/n
xiR = a + i∆x
1679251462458.png


Hi! I am having trouble finalizing this problem.

The interval is given so we know that a = 1 and b = 2. From there you can figure out that ∆x = 1/n, xiR = 1 + i/n.
Using logarithmic properties, I rearranged the expression and wrote (1 + i/n)(1/n)ln[(n + i)/n].
I can guess that the function is going to look something like this: f(x) = xln(...) but I don't know what goes in the logarithmic function...

I have been trying to rewrite it in terms of xiR but no luck :(
any help would be really appreciated. thank you !
 
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  • #2
Can you simplify the argument of the logarithm any further?
 
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Likes crememars
  • #3
pasmith said:
Can you simplify the argument of the logarithm any further?
Yes, I realized I was overcomplicating things. If I simplify (n+i)/n to (n/n +i/n) to (1+ i/n), I get f(x) = xlnx
 

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