To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $$U ( f ) ( x ) = \int _ { 0 } ^ { x } \left( 1 - \mathrm { e } ^ { - t } \right) \frac { f ( t ) } { t } \mathrm {~d} t + \left( \mathrm { e } ^ { x } - 1 \right) \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$ Let $f \in E$. Show that $U ( f )$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$ and satisfies, for all $x > 0$, $$( U ( f ) ) ^ { \prime } ( x ) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$
To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by
$$U ( f ) ( x ) = \int _ { 0 } ^ { x } \left( 1 - \mathrm { e } ^ { - t } \right) \frac { f ( t ) } { t } \mathrm {~d} t + \left( \mathrm { e } ^ { x } - 1 \right) \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$
Let $f \in E$. Show that $U ( f )$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$ and satisfies, for all $x > 0$,
$$( U ( f ) ) ^ { \prime } ( x ) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$