Chapter 5.10, Problem 16E

To determine

To find whether the integral ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\cos \pi tdt}$ is convergent or divergent.

Answer to Problem 16E

The integral ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\cos \pi tdt}$ is divergent.

Explanation of Solution

Given information:

The integral is ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\cos \pi tdt}$

Concept used: If the value of the integral is finite, it is convergent otherwise it is divergent

Given, ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\cos \pi tdt}$

Let $f(x)=\cos \pi t$ , Which is an even function

The given integral ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f(x)dx=2{\displaystyle \underset{0}{\overset{\infty }{\int }}f(x)dx}}$

Thus, given integral ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\cos \pi tdt}=2{\displaystyle \underset{0}{\overset{\infty }{\int }}\cos \pi tdt}$

On integration, the integral becomes as:

  $=2{[\frac{\sin \pi t}{\pi }]}_0^{\infty },\text{which does not exist [sin}\infty \text{ does not converge to a point]}$

Since the given integral does not exist.

Hence, the given integral is divergent