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Let's go over the exercises one by one.

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### Exercice 2
We are asked to compute the integral:

\[
I = \int_0^\pi \frac{t \sin(t)}{1 + \cos^2(t)} dt
\]

To approach this, we could attempt integration by parts or explore symmetry and substitution techniques to simplify the expression. It may require some trigonometric identities to handle the integral in a more manageable form.

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### Exercice 3
The goal is to calculate the integral:

\[
I = \int_0^1 e^{-x} \ln(1 + e^x) dx
\]

This integral involves the combination of an

Exercice 2: Calculez I = ∫_0^π (t * sin(t)) / (1 + cos^2(t)) dt, Exercice 3: Calculez I = ∫_0^1 e^(-x) ln(1 + e^x) dx, Exercice 4: Soit n ∈ N*. On définit I_n = ∫_0^(π/2) sin^n(t) dt. Trouver une égalité entre I_n+2 et I_n.

This page explores the solutions to advanced integral problems involving trigonometric functions, logarithms, and recursive relations in integrals. Problems include calculating integrals like I = ∫_0^π (t * sin(t)) / (1 + cos^2(t)) dt, I = ∫_0^1 e^(-x) ln(1 + e^x) dx, and finding a relationship between I_n+2 and I_n for sin^n(t) integrals.

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