17-20 Express the limit as a definite integral on the given interval.
17. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{e^{x_{i}}}{1+x_{i}} \Delta x,[0,1]$

Answer

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Answer

The limit is expressed as the definite integral $\int_{0}^{1} \frac{e^{x}}{1+x} dx$.

Steps

Step 1 ：The given limit is a Riemann sum for the function $f(x) = \frac{e^{x}}{1+x}$ on the interval $[0,1]$.

Step 2 ：The limit of this sum as $n$ approaches infinity is the definite integral of $f(x)$ from 0 to 1.

Step 3 ：Therefore, we need to calculate the definite integral of $f(x)$ from 0 to 1.

Step 4 ：The limit is expressed as the definite integral $\int_{0}^{1} \frac{e^{x}}{1+x} dx$.