The integral in this exercise converges. Evaluate the integral without using a table. \( \int_{-\infty}^{\infty} \frac{2 x d x}{\left(x^{2}+3\right)^{2}} \) \( \int_{-\infty}^{\infty} \frac{2 x d x}{\left(x^{2}+3\right)^{2}}=\square \)

To evaluate the integral \( \int_{-\infty}^{\infty} \frac{2 x \, dx}{(x^{2}+3)^{2}} \), we can notice that the integrand is an odd function. An odd function has the property that \( f(-x) = -f(x) \). Since the limits of integration are symmetric about the origin, the integral of an odd function over a symmetric interval around zero evaluates to zero. Thus, \[ \int_{-\infty}^{\infty} \frac{2 x \, dx}{(x^{2}+3)^{2}} = 0. \] Remember, integrals can lead to interesting symmetries in mathematics! This property can often help simplify problems without heavy lifting and heavy calculations. If you’re ever puzzled, take a moment to consider whether the function you're dealing with might have a useful symmetry or property to exploit. A seemingly challenging integral might turn into an effortless exercise simply because it’s odd or even!