int_{-frac{pi}{8092}}^{frac{pi}{8092}} frac{s | vedclass

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Question

(B) Let $I = \int_{-\frac{\pi}{8092}}^{\frac{\pi}{8092}} \frac{\sec (2023 x)}{1+(2023)^{(2023 x)}} d x$. Using the property $\int_{-a}^{a} f(x) d x =

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$\frac{\log (\sqrt{2}+1)}{2023}$

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(B) Let $I = \int_{-\frac{\pi}{8092}}^{\frac{\pi}{8092}} \frac{\sec (2023 x)}{1+(2023)^{(2023 x)}} d x$. Using the property $\int_{-a}^{a} f(x) d x = \int_{0}^{a} [f(x) + f(-x)] d x$, we have: $f(x) = \frac{\sec (2023 x)}{1+(2023)^{(2023 x)}}$ $f(-x) = \frac{\sec (-2023 x)}{1+(2023)^{(-2023 x)}} = \frac{\sec (2023 x)}{1+\frac{1}{(2023)^{(2023 x)}}} = \frac{(2023)^{(2023 x)} \sec (2023 x)}{(2023)^{(2023 x)}+1}$ Adding $f(x)$ and $f(-x)$: $f(x) + f(-x) = \frac{\sec (2023 x) [1 + (2023)^{(2023 x)}]}{1+(2023)^{(2023 x)}} = \sec (2023 x)$ Therefore, $I = \int_{0}^{\frac{\pi}{8092}} \sec (2023 x) d x$ $= \left[ \frac{1}{2023} \log |\sec (2023 x) + 	an (2023 x)| ight]_{0}^{\frac{\pi}{8092}}$ $= \frac{1}{2023} [\log |\sec(\frac{\pi}{4}) + 	an(\frac{\pi}{4})| - \log |\sec(0) + 	an(0)|]$ $= \frac{1}{2023} [\log (\sqrt{2} + 1) - \log (1 + 0)]$ $= \frac{\log (\sqrt{2} + 1)}{2023}$

$\int_{-\frac{\pi}{8092}}^{\frac{\pi}{8092}} \frac{\sec (2023 x)}{1+(2023)^{(2023 x)}} d x=$

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