Can you handle this integration with limits problem?

[Vaibhav Dixit 1008]

Homework Statement

Integrate ∫ (tan √x) / (2 √x) dx

Homework Equations

Limits from 0 to ∞

The Attempt at a Solution

Put u = √x
du/dx = 1/ (2 √x)
dx = du * (2 √x)
now question becomes
∫ tan u du = log sec u = log (sec √x)
now applying limits
∫ tan u du = log (sec √∞) - log (sec √0)
= log (sec √∞)
Now what is this
I' m not getting

 

[Ray Vickson]

Homework Statement

Integrate ∫ (tan √x) / (2 √x) dx

Homework Equations

Limits from 0 to ∞

The Attempt at a Solution

Put u = √x
du/dx = 1/ (2 √x)
dx = du * (2 √x)
now question becomes
∫ tan u du = log sec u = log (sec √x)
now applying limits
∫ tan u du = log (sec √∞) - log (sec √0)
= log (sec √∞)
Now what is this
I' m not getting

Have you analyzed whether or not the integral
$$I = \int_0^{\infty} \frac{\tan \sqrt{x}}{2 \sqrt{x}} \, dx$$
is convergent?

 

[MidgetDwarf]

Homework Statement

Integrate ∫ (tan √x) / (2 √x) dx

Homework Equations

Limits from 0 to ∞

The Attempt at a Solution

Put u = √x
du/dx = 1/ (2 √x)
dx = du * (2 √x)
now question becomes
∫ tan u du = log sec u = log (sec √x)
now applying limits
∫ tan u du = log (sec √∞) - log (sec √0)
= log (sec √∞)
Now what is this
I' m not getting

You have to play with the limits of integration. Notice that lower limit of integration is 0. Does this cause a problem? What if we "changed" the lower limit of integration somewhat. i.e. Let t=0. Then take the limit as t approaches 0 from the right... Does this ring any bells?

 

[Mark44]

Aside from whatever problem there might be right around x = 0, there are an infinite number of serious discontinuities for the function $f(x) = \frac {\tan(\sqrt(x)}{2\sqrt(x)}$