Let’s break down the exercises in this image step by step for clarity and solutions:

Exercice 1: Calculating Limits using Riemann Sums

We are tasked to calculate three limits by interpreting them as Riemann sums:

1. $\lim_{n \to \infty} \left( \sum_{k=1}^n \frac{1}{n^2 + k^2} \right)$

2. $\lim_{n \to \infty} \sum_{k=1}^n \frac{k}{n^2 + k^2}$

3. $\lim_{n \to \infty} \sum_{k=1}^{n-1} \frac{k}{\sqrt{n^2 + k^2}}$

For each, we will:

• Convert the summation into a Riemann sum.
• Identify the corresponding integral.
• Solve the integral.

Exercice 2: Linear Algebra

1. Show that $E_1 = \{ (x, y, z) \in \mathbb{R}^3 : f(x, y, z) = (x, y, z) \}$ is a subspace of $\mathbb{R}^3$:

   • Prove $E_1$ is closed under addition and scalar multiplication.
   • Demonstrate it forms a subspace.

2. Given $b = (0, 1, 1)$ and $c = (1, 4, -1)$:

   • Compute their image under the transformation $f$.
   • Show their independence using the determinant or by checking the rank of their matrix.

3. Demonstrate that $B' = (a, b, c)$ forms a basis of $\mathbb{R}^3$:

   • Use independence or determinant calculations to verify.

4. Find the change of basis matrix $P$ from $B$ to $B'$:

   • Express $e_1, e_2, e_3$ (canonical basis) in terms of $B'$.

5. Compute the inverse of $P$:

   • Use the standard formula for matrix inversion.

Exercice 3: Variable Substitution in Integrals

1. $I = \int_0^{\ln 2} \sqrt{1 + e^x} dx, \quad \text{with } t = \sqrt{1 + e^x}$

   • Perform substitution to rewrite the integral in terms of $t$.

2. $J = \int_0^1 \frac{\sqrt{1 - x^2}}{1 + x^2} dx, \quad \text{with } t = \cos x$

   • Apply the substitution $t = \cos x$.

3. $K = \int_0^1 \frac{1}{1 + \sqrt{1+x}} dx, \quad \text{with } t = 1 + \sqrt{1+x}$

   • Simplify the integral with the given substitution.

Would you like me to begin solving these problems step by step or prioritize a specific one?