It looks like you've uploaded an image containing a math problem in French. Let's break it down:

Part a)

We are asked to calculate the sum: $\sum_{k=1}^{n} I_k$ where $I_k$ is given by the following integral: $I_k = \int_0^1 C_n^k x^k (1 - x)^{n-k} \, dx$ and $C_n^k$ is the binomial coefficient.

Part b)

Here, we need to find a relation between $I_k$ and $I_{k+1}$ using integration by parts.

Part c)

Finally, we need to express $I_k$ as a function of $n$.

I'll go ahead and start with part a. First, the integral expression for $I_k$ looks like a typical binomial expansion, so we might be able to simplify it using properties of binomial coefficients.

Would you like me to proceed with a detailed solution for part a? Or do you need any specific clarifications about the problem?