Chapter 7.7, Problem 100E

To determine

(a)

To show:

${\displaystyle \underset{1}{\overset{R}{\int }}\frac{\sin x}{x}dx=-{\frac{\cos x}{x}|}_1^R-{\displaystyle \underset{1}{\overset{R}{\int }}\frac{\cos x}{x^2}dx}}$

To determine

(b)

To show:

${\displaystyle \underset{1}{\overset{\infty }{\int }}\frac{\cos x}{x^2}dx}$ converges. Also conclude that the limit as $R\to \infty$ of the integral $(a)$ exists and is finite.

To determine

(c)

To show that ${\displaystyle \underset{0}{\overset{\infty }{\int }}f\left(x\right)dx}$ converges.