Q2
Integration by Parts
Differentiation Under the Integral Sign Combined with Parts
View
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 $p > q > 0$.
$$\int _ { 0 } ^ { \infty } \frac { e ^ { - p x } \cos ( p x ) - e ^ { - q x } \cos ( q x ) } { x } \mathrm {~d} x$$