i.2.32 Integrals depending on a parameter
condition: Prove that \[ \int_{0}^{+\infty} e^{-4} d x \cdot \int_{0}^{+\infty} x^{2} e^{-x^{4}} d x=\frac{\pi}{8 \sqrt{2}} \]
Integrals depending on a parameter, often referred to as parameter-dependent integrals or definite integrals with parameters, involve the integration of a function where one or more parameters are present. These parameter-dependent integrals can lead to a variety of problems and challenges. Here you can find problems related to convergence and divergence, calculation, limits, parametric integration and differentiation of parameter-dependent integrals. Also, there are usage and properties of Euler, Frullani, Poisson, Laplace, and Dirichlet integrals.