Problem 1: calculation of an integral For $x \geqslant 0$ we define $$f ( x ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - t x } } { 1 + t ^ { 2 } } \mathrm {~d} t \quad \text { and } \quad g ( x ) = \int _ { 0 } ^ { \infty } \frac { \sin t } { t + x } \mathrm {~d} t$$ Study of $g$. a. Show that $g ( x )$ is well defined for all $x \geqslant 0$. b. Show that $g$ is of class $C ^ { 2 }$ on $] 0 , \infty [$. For this you may use the change of variable $u = t + x$ and express $g$ in terms of the functions $C : x \mapsto \int _ { x } ^ { \infty } \frac { \cos u } { u } \mathrm {~d} u$ and $S : x \mapsto \int _ { x } ^ { \infty } \frac { \sin u } { u } \mathrm {~d} u$. c. Determine a linear second-order differential equation satisfied by $f$.
\textbf{Problem 1: calculation of an integral}
For $x \geqslant 0$ we define
$$f ( x ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - t x } } { 1 + t ^ { 2 } } \mathrm {~d} t \quad \text { and } \quad g ( x ) = \int _ { 0 } ^ { \infty } \frac { \sin t } { t + x } \mathrm {~d} t$$
Study of $g$.\\
a. Show that $g ( x )$ is well defined for all $x \geqslant 0$.\\
b. Show that $g$ is of class $C ^ { 2 }$ on $] 0 , \infty [$.
For this you may use the change of variable $u = t + x$ and express $g$ in terms of the functions $C : x \mapsto \int _ { x } ^ { \infty } \frac { \cos u } { u } \mathrm {~d} u$ and $S : x \mapsto \int _ { x } ^ { \infty } \frac { \sin u } { u } \mathrm {~d} u$.\\
c. Determine a linear second-order differential equation satisfied by $f$.