To calculate the integrals provided in the image, let's tackle them one by one. Here's the breakdown of the given integrals:

1. $I = \int_0^{\ln a} (1 + e^x) \ln(x + e^x) \, dx$, with $a \in [1, +\infty[$.
2. $J = \int_{\ln \frac{\pi}{4}}^{\ln \frac{\pi}{2}} e^{2x} \sin(e^x) \, dx$.
3. $K = \int_0^{\sqrt{3}} \frac{x^3}{\sqrt{1+x^2}} \, dx$.
4. $L = \int_0^{\frac{\pi}{2}} \cos x \ln(1 + \cos x) \, dx$.
5. $M = \int_1^2 \frac{1}{x^3} e^{\frac{1}{x}} \, dx$.
6. $N = \int_0^{2\pi} \frac{\ln(1+x)}{\sqrt{1+x}} \, dx$.

I will solve them systematically. Let me start with $I$. sympy import symbols, exp, ln, integrate, sqrt, sin, cos, pi

Define the variables

x, a = symbols('x a', positive=True)

Integral I

I_expr = (1 + exp(x)) * ln(x + exp(x)) I = integrate(I_expr, (x, 0, ln(a))) IThe