To calculate the integral J = integral ((x)(x))/(^2(x)+1) dx, we can use a substitution method. The problem statement mentions "règle de Bioche" (Bioche's rule), which provides guideline

To calculate the integral $J = \int \frac{\sin(x)\cos(x)}{\sin^2(x)+1} dx$, we can use a substitution method. The problem statement mentions "règle de Bioche" (Bioche's rule), which provides guidelines for substitutions in integrals involving trigonometric functions.

Let the integrand be $\omega(x) = \frac{\sin(x)\cos(x)}{\sin^2(x)+1}$. We will check the conditions given by Bioche's rule:

1. Check if $\omega(x) = \omega(-x)$: $\omega(-x) = \frac{\sin(-x)\cos(-x)}{\sin^2(-x)+1} = \frac{-\sin(x)\