Definite integral approaches infinity

[JasonHathaway]

Homework Statement

180\int_5^\propto \frac{2}{(4+x^{2})^{3/2}} dx

Homework Equations

Trigonometric Substitutions: (x=2 tan z).

The Attempt at a Solution

I've computed the integral and ended up with 180 [\frac{x}{2(4+x^2)^{1/2}}] from 5 to infinity.

I could've easily computed the limit of 5, but couldn't have found the limit approaches infinity, as I'm always ending up with 0, although my textbook and symolab calculator says that it's 1/2.

 

[Ray Vickson]

Homework Statement

180\int_5^\propto \frac{2}{(4+x^{2})^{3/2}} dx

Homework Equations

Trigonometric Substitutions: (x=2 tan z).

The Attempt at a Solution

I've computed the integral and ended up with 180 [\frac{x}{2(4+x^2)^{1/2}}] from 5 to infinity.

I could've easily computed the limit of 5, but couldn't have found the limit approaches infinity, as I'm always ending up with 0, although my textbook and symolab calculator says that it's 1/2.

Either use l'Hospital's rule or else note that for x > 0 we have
\frac{x}{\sqrt{x^2+4}} = \frac{x}{x \sqrt{1 + 4x^{-2}}} = \frac{1}{\sqrt{1+4 x^{-2}}} \to 1
as $x \to \infty$.

 

[SteamKing]

\propto means 'is proportional to' and should not be used to indicate infinity \infty

 

[infinitylord]

If it is infinity and not saying proportional to. Then you can calculate the limit to infinity pretty easily.

Take \frac{x}{2\sqrt{4+x^{2}}} and then make the numerator \sqrt{x^{2}} and pull out the \frac{1}{2}

So now you can call the whole thing \frac{1}{2}\sqrt{\frac{x^{2}}{4+x^{2}}}

Now apply L'Hopital's rule.

\frac{1}{2}\sqrt{\frac{2x}{2x}}

this way, the 2x and 2x cancel to give you \sqrt{1} multiplied by \frac{1}{2}. Which was your answer.