Determine if the improper integral converges or diverges

[physics=world]

1. integrate from (1 to 3) of function (2) / (x-2)^(8/3)

Can someone explain why this diverges. i do not understand it. when i plugged in the numbers there are no discontinuities and this is where i am stuck at. If there are no discontinuity does that means that it diverges?

Homework Equations

The Attempt at a Solution

 

[jbunniii]

1. integrate from (1 to 3) of function (2) / (x-2)^(8/3)

Can someone explain why this diverges. i do not understand it. when i plugged in the numbers there are no discontinuities and this is where i am stuck at. If there are no discontinuity does that means that it diverges?

What happens at $x=2$?

 

[physics=world]

oh. it would equal to zero. so does that mean that it is continuous on the interval [1,3] except at 2? if so, do i proceed with solving it from 1,2 to 2,3 ?

 

[iRaid]

Solve it from 1 to t, t to 3 and do the limit as t approaches 2 from the right and left.

 

[physics=world]

i got the answer -12/5. Since its negative does that means that it diverges?

 

[lordsurya08]

Graph the function in your head...as it approaches 2 the denominator (x-2) term goes to zero, so the function goes to infinity. Hence the area under the curve also goes to infininity (diverges).

 

[jbunniii]

Graph the function in your head...as it approaches 2 the denominator (x-2) term goes to zero, so the function goes to infinity. Hence the area under the curve also goes to infininity (diverges).

That's not automatically true. For example, $1/|x|^{1/2}$ diverges to infinity as $x \rightarrow 0$, but the function has a finite (improper) integral over any finite-length interval even if the interval includes 0.

In general, whether the integral diverges or not at a singularity depends on how "wide" the singularity is: the integral of $1/|x|^p$ over an interval including 0 will converge or diverge depending on the value of $p$. Larger $p$ = wider singularity.

 

[jbunniii]

i got the answer -12/5. Since its negative does that means that it diverges?

If you got a finite answer (positive or negative), that would mean the integral converges. However, please check your work or post it here. -12/5 is incorrect.