MHB Calculate Integral: $\int_0^1 \frac{(1-x)e^x}{x+e^x}dx$

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The integral $\int_0^1 \frac{(1-x)e^x}{x+e^x}dx$ presents challenges that do not align with standard calculus techniques. Suggestions for solving it include exploring contour integration, differentiation under the integral sign, or applying Laplace transforms. Participants express uncertainty about its origin and the most effective method for evaluation. The integral's complexity indicates it may require advanced mathematical approaches. Ultimately, finding a solution may involve innovative strategies beyond traditional calculus methods.
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Calculate the following integral
$$\int\limits_0^1\frac{(1-x)e^x}{x+e^x}dx$$
 
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From what problem did this integral arise? It doesn't seem to succumb to any standard techniques in regular calculus. You might be able to try contour integration, or maybe differentiation under the integral sign. Or maybe even Laplace Transforms.
 

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