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Question Number 208242 by alcohol last updated on 08/Jun/24

Commented by alcohol last updated on 08/Jun/24

$p l e a s e h e l p m e t r a n s l a t e a n d s o l v e$
	Answered by mr W last updated on 09/Jun/24

	$\int_{a}^{b} f (x) d x$ $= \underset{k = 0}{\sum^{n - 1}} \int_{x_{2 k}}^{x_{2 (k + 1)}} f (x) d x$ $\approx \underset{k = 0}{\sum^{n - 1}} \frac{x_{2 (k + 1)} - x_{2 k}}{6} [y_{2 k} + 4 y_{2 k + 1} + y_{2 (k + 1)}]$ $\approx \frac{b - a}{6 n} \underset{k = 0}{\sum^{n - 1}} [y_{2 k} + 4 y_{2 k + 1} + y_{2 (k + 1)}]$ $\approx \frac{b - a}{6 n} [2 (y_{0} + y_{2} + y_{4} + . . . + y_{2 n}) + 4 (y_{1} + y_{3} + y_{5} + . . . + y_{2 n - 1}) - y_{0} - y_{2 n}]$ $\approx \frac{b - a}{6 n} [y_{0} + y_{2 n} + 2 (y_{2} + y_{4} + . . . + y_{2 n - 2}) + 4 (y_{1} + y_{3} + y_{5} + . . . + y_{2 n - 1})]$ $\ln 2 = \int_{1}^{2} \frac{d x}{x}$ $\approx \frac{2 - 1}{6 \times 2} [\frac{1}{1} + \frac{1}{2} + 2 (\frac{4}{6}) + 4 (\frac{4}{5} + \frac{4}{7})]$ $\approx 0.69325 (w i t h n = 2)$ $o r$ $\approx \frac{2 - 1}{6 \times 3} [\frac{1}{1} + \frac{1}{2} + 2 (\frac{6}{8} + \frac{6}{10}) + 4 (\frac{6}{7} + \frac{6}{9} + \frac{6}{11})]$ $\approx 0.69317 (w i t h n = 3)$
	Commented by mr W last updated on 09/Jun/24

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