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A sequence \((u_n)\) is defined for any integer \(n > 0\) by the integral:$$ u_n = \int_0^1 \frac{e^{nx}}{1+e^x} \,dx $$
  1. Calculate \(u_1\).
  2. Prove that for any integer \(n>0\), the following recurrence relation holds: $$u_{n+1} + u_n = \frac{e^n - 1}{n}$$
  3. Hence, deduce the value of \(u_2\).

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