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 $f$. a. Show that $f ( x )$ is well defined for all $x \geqslant 0$. b. Show with precision that the function $f$ is of class $C ^ { 2 }$ on $] 0 , \infty [$, and also continuous at 0. c. Calculate $f + f ^ { \prime \prime }$ and deduce that $f$ is a solution of a linear second-order differential equation.
\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 $f$.\\
a. Show that $f ( x )$ is well defined for all $x \geqslant 0$.\\
b. Show with precision that the function $f$ is of class $C ^ { 2 }$ on $] 0 , \infty [$, and also continuous at 0.\\
c. Calculate $f + f ^ { \prime \prime }$ and deduce that $f$ is a solution of a linear second-order differential equation.