Question

How do you find `\int _ { 0} ^ { + \infty } \frac { 1} { \sqrt { z ^ { 2} + R ^ { 2} } } d z`?

Answer 1

Does not converge

Explanation:

For the indefinite integral

`\int\frac { 1} { \sqrt { z ^ { 2} + R ^ { 2} } } d z`

`= 1/R \int \frac { 1} { \sqrt { (z/R) ^ { 2} + 1 } } d z`

let `z/R = sinh alpha, dz = R cosh alpha \ d alpha`

The integral becomes:

`1/R \int \frac { 1} { \sqrt { sinh^ { 2} alpha + 1 } } * R cosh alpha \ d alpha`

`= \int d alpha = alpha + C = sinh^(-1) (z/R) + C`

That leaves the definite integral as:

`lim_( beta to oo) [sinh^(-1) (z/R)]_0^beta`

This does not converge: `lim_(x to oo) sinh^(-1) x = oo`

This has not considered any possible restrictions on the value of `R`, as that wouldn't seem to affect the conclusion drawn.

graph{sinh x [-10, 10, -5, 5]}