1. ## Improper Integrals

I'm having trouble starting the following problem...

Explain why the integral is improper.

$\int_1^4 \frac{dx}{x^2 lnx}$

2. Because ln(1)=0 making the integrand have a discontinuity at x=1

3. Originally Posted by Mathstud28
Because ln(1)=0 making the integrand have a discontinuity at x=1
Wouldn't that just mean that you subtract that 0 from the function when 4 is plugged in? This may sound a little confusing.

4. ## and If I am not mistaken

all you do is take the integral except make the limits of integration lim[b as b approaches 1] and 4 and if it still doesnt work it diverges

5. ## Haha it does

but...hmmm If I think what you are saying is what you are really saying then the answer is no...you cant just throw out an aspect in lieu of an easier integral

6. Originally Posted by larson
Wouldn't that just mean that you subtract that 0 from the function when 4 is plugged in? This may sound a little confusing.
You need to read the first few lines of this before you'll understand the answer to your question.

7. Ok so another easy way to tell if it is improper is to just graph the function and see if you get any horizontal lines (aka errors) within the graph.

8. Originally Posted by larson
Ok so another easy way to tell if it is improper is to just graph the function and see if you get any horizontal lines (aka errors) within the graph.
No. Improper if the integrand has vertical asymptotes in the interval of integration.

9. Originally Posted by mr fantastic
No. Improper if the integrand has vertical asymptotes in the interval of integration.
Errr... thats what I meant :\ sorry. Thanks though for the correction.

10. ## IT is like I said

if the integrand (The thing you are taking the integral of) has a discontinuity that is an element of [a,b] where a and be are the limits of integration then the integral is improper

11. Originally Posted by Mathstud28
if the integrand (The thing you are taking the integral of) has a discontinuity that is an element of [a,b] where a and be are the limits of integration then the integral is improper
No. Consider $f(x) = \left\{ \begin{array}{c}-1 \mbox{ for }-1\leq x\leq 0 \\ 1\mbox{ for }0 then $\int_{-1}^1 f(x) dx$ is not an improper integral.

12. Originally Posted by Mathstud28
if the integrand (The thing you are taking the integral of) has a discontinuity that is an element of [a,b] where a and be are the limits of integration then the integral is improper
What TPH is true. Note that not all discontinuities are vertical asymptotes ......