Chapter 8.8, Problem 90E

(a)

To determine

The explanation of the difference between the integrands ${\displaystyle {\int }_1^{\infty }\frac{1}{x}dx}$ and ${\displaystyle {\int }_1^{\infty }\frac{1}{x^2}dx}$ which cause one integral to converge and other to diverge.

If the integrands ${\displaystyle {\int }_1^{\infty }\frac{1}{x}dx}$ and ${\displaystyle {\int }_1^{\infty }\frac{1}{x^2}dx}$ causes an integral to converge and the other to diverge. Explain the differences between the two integrands.

(b)

To determine

To graph: The function $y=\frac{\left(\sin x\right)}{x}$ is convergent or not overaninterval $\left(1,\infty \right)$.

(c)

To determine

To calculate: The Convergence of integrand ${\displaystyle {\int }_1^{\infty }\frac{\sin x}{x}dx}$ using by parts.