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. Participants in the discussion suggest advanced methods such as contour integration, differentiation under the integral sign, and Laplace Transforms as potential solutions. These approaches indicate the integral's complexity and the necessity for higher-level mathematical tools to evaluate it effectively.

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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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