Chapter A, Problem 53E

(a)

To determine

To show: The integral value of ${\displaystyle {\int }_0^{\infty }\frac{1}{x^2+t^2}}dt$ is $\frac{\pi }{2x}$.

(b)

To determine

To show: The integral value of ${\displaystyle {\int }_0^{\infty }\frac{1}{{\left(x^2+t^2\right)}^2}}dt$ is $\frac{\pi }{4x^3}$ by the use of ${\displaystyle {\int }_0^{\infty }\frac{1}{x^2+t^2}}dt=\frac{\pi }{2x}$.