6. [-/2 Points $]$ DETAILS SCALC9 4.2.022.
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Express the limit as a definite integral on the given interval.
\[
\begin{array}{l}
\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{x_{i}^{*}}{\left(x_{i}^{*}\right)^{2}+1} \Delta x, \quad[1,8] \\
\int_{1}(\square d x \\
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\end{array}
\]
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Answer
The definite integral of the function on the interval [1, 8] is \(\boxed{-\frac{\log(2)}{2} + \frac{\log(65)}{2}}\).
Step 1 ：The question is asking to express the limit of the sum as a definite integral. The limit of the sum is a Riemann sum which is the definition of a definite integral.
Step 2 ：The function inside the sum is \(f(x) = \frac{x}{x^2 + 1}\) and the interval is [1, 8].
Step 3 ：Therefore, the limit of the sum can be expressed as the definite integral of \(f(x)\) from 1 to 8.
Step 4 ：The definite integral of the function on the interval [1, 8] is \(\boxed{-\frac{\log(2)}{2} + \frac{\log(65)}{2}}\).
