For a real number $t$, define the function $f ( x )$ as $$f ( x ) = \left\{ \begin{array} { c c } 1 - | x - t | & ( | x - t | \leq 1 ) \\ 0 & ( | x - t | > 1 ) \end{array} \right.$$ For a certain odd number $k$, the function $$g ( t ) = \int _ { k } ^ { k + 8 } f ( x ) \cos ( \pi x ) d x$$ satisfies the following condition.
When all $\alpha$ for which the function $g ( t )$ has a local minimum at $t = \alpha$ and $g ( \alpha ) < 0$ are listed in increasing order as $\alpha _ { 1 } , \alpha _ { 2 } , \cdots , \alpha _ { m }$ (where $m$ is a natural number), we have $\sum _ { i = 1 } ^ { m } \alpha _ { i } = 45$. Find the value of $k - \pi ^ { 2 } \sum _ { i = 1 } ^ { m } g \left( \alpha _ { i } \right)$. [4 points]

Consider the following integral $I _ { n } ( \alpha )$ for $\alpha \geq 1$ and $n > 0$.
$$I _ { n } ( \alpha ) = \int _ { \frac { 1 } { n } } ^ { n } \frac { f ( \alpha x ) - f ( x ) } { x } \mathrm {~d} x$$
Assume that a real-valued function $f ( x )$ is continuous and differentiable on $x \geq 0$, its derivative is continuous, and $\lim _ { x \rightarrow \infty } f ( x ) = 0$. Answer the following questions.
(1) Define $J _ { n } ( \alpha ) = \frac { \mathrm { d } I _ { n } ( \alpha ) } { \mathrm { d } \alpha }$. Show that $J _ { n } ( \alpha ) = \frac { 1 } { \alpha } \left( f ( \alpha n ) - f \left( \frac { \alpha } { n } \right) \right)$.
You can use the fact that the integration and the differentiation commute in this context.
(2) Define $I ( \alpha ) = \lim _ { n \rightarrow \infty } I _ { n } ( \alpha )$. Show that $\lim _ { n \rightarrow \infty } J _ { n } ( \beta )$ exists for any $\beta \in [ 1 , \alpha ]$ and it uniformly converges on $[ 1 , \alpha ]$, and show that
$$I ( \alpha ) = \int _ { 1 } ^ { \alpha } \left( \lim _ { n \rightarrow \infty } J _ { n } ( \beta ) \right) \mathrm { d } \beta$$
(3) Obtain $I ( \alpha )$.
(4) Calculate the following integral. Note that $p > q > 0$.
$$\int _ { 0 } ^ { \infty } \frac { e ^ { - p x } \cos ( p x ) - e ^ { - q x } \cos ( q x ) } { x } \mathrm {~d} x$$

The functions $f$ and $g$ are defined and differentiable on the set of real numbers and satisfy
$$\begin{aligned} & \int_{1}^{2} f^{\prime}(3x)\, dx = 4 \\ & \int f(2x)\, dx = g(x) + C, \quad (C \text{ constant}) \end{aligned}$$
If $f(3) = 5$, what is the value of the derivative $g^{\prime}(3)$?
A) 1 B) 5 C) 9 D) 13 E) 17