Consider the following integral:
. . . . .\(\displaystyle \displaystyle \int_2^6\, \dfrac{x}{1\, +\, x^5}\, dx\)
Which of the following expressions represents the integral as a limit of Riemann sums?
. . .\(\displaystyle \displaystyle \mbox{A. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{4}{n}\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)^5}\)
. . .\(\displaystyle \displaystyle \mbox{B. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{4}{n}\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)}\)
. . .\(\displaystyle \displaystyle \mbox{C. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{6}{n}\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)^5}\)
. . .\(\displaystyle \displaystyle \mbox{D. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)^5}\)
. . .\(\displaystyle \displaystyle \mbox{E. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{6}{n}\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)}\)
. . .\(\displaystyle \displaystyle \mbox{F. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)^5}\)
. . . . .\(\displaystyle \displaystyle \int_2^6\, \dfrac{x}{1\, +\, x^5}\, dx\)
Which of the following expressions represents the integral as a limit of Riemann sums?
. . .\(\displaystyle \displaystyle \mbox{A. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{4}{n}\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)^5}\)
. . .\(\displaystyle \displaystyle \mbox{B. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{4}{n}\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)}\)
. . .\(\displaystyle \displaystyle \mbox{C. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{6}{n}\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)^5}\)
. . .\(\displaystyle \displaystyle \mbox{D. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)^5}\)
. . .\(\displaystyle \displaystyle \mbox{E. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{6}{n}\, \dfrac{2\, +\, \frac{6i}{n}}{1\, +\, \left(2\, +\, \frac{6i}{n}\right)}\)
. . .\(\displaystyle \displaystyle \mbox{F. }\, \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, \dfrac{2\, +\, \frac{4i}{n}}{1\, +\, \left(2\, +\, \frac{4i}{n}\right)^5}\)
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