$\bf命题：$讨论$\quad$$I\left( y \right) = \int_0^{ + \infty } {\frac{{\sin {x^2}}}{{1 + {x^y}}}dx} $$\quad$在$\left[ {0, + \infty } \right)$上的一致收敛性
$\bf命题：$
$\bf(10中南大学六)$已知$\int_0^{ + \infty } {\frac{{\sin \pi x}}{x}dx = \frac{\pi }{2}} $，求$I\left( a \right) = \int_0^{ + \infty } {{e^{ - ax}}\frac{{\sin \pi x}}{x}dx} \left( {a > 0} \right)$
$\bf(12川大七)$设$f\left( x \right) = \int_1^{ + \infty } {\frac{{\sin xt}}{{t\left( {1 + {t^2}} \right)}}} dt,x \in \left( { - \infty , + \infty } \right)$
(1)证明：$f(x)$关于$x$在$( - \infty , + \infty)$上一致收敛
(2)证明：$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty } \end{array}} f\left( x \right) = 0$
(3)证明：$f(x)$在$( - \infty , + \infty)$上一致连续