Convergence of Improper Integrals: Two Examples

[clairez93]

Homework Statement

Determine the divergence or convergence of the improper integral. Evaluate the integral if it converges.

1. \int^{4}_{2}\frac{1}{\sqrt{x^{2}-4}} dx

2. \int^{2}_{0}\frac{1}{\sqrt[3]{x-1}} dx

Homework Equations

The Attempt at a Solution

1.

t = x - 2
dt = dx

\int^{4}_{2}\frac{1}{\sqrt{x^{2}-4}} dx
=\int^{2}_{0}\frac{1}{\sqrt{(t+2)^{2}-4}} dx
=\int^{2}_{0}\frac{1}{\sqrt{t^{2}+4t}} dx

\stackrel{lim}{x\rightarrow\infty} \int^{2}_{N}\frac{1}{\sqrt{t^{2}+4t}} dt

I'm not sure how to integrate \int^{2}_{N}\frac{1}{\sqrt{t^{2}+4t}} dt from here.


2.

\int^{2}_{0}\frac{1}{\sqrt[3]{x-1}} dx = \int^{1}_{0}\frac{1}{\sqrt[3]{x-1}} + \int^{2}_{1}\frac{1}{\sqrt[3]{x-1}}

t = 1 - x
dt = -dx

\int^{1}_{0}\frac{1}{\sqrt[3]{x-1}} = \int^{0}_{1}\frac{-1}{\sqrt[3]{-t}} dt = \int^{1}_{0}\frac{1}{\sqrt[3]{-t}} dt


\stackrel{lim}{x\rightarrow\infty} \int^{1}_{N}-t^{-1/3} dt
= \stackrel{lim}{x\rightarrow\infty} -\frac{3}{4}(-t)^{4/3}|^{1}_{N} = \stackrel{lim}{x\rightarrow\infty} \left[\frac{3}{4} + \frac{3}{4}(-N)^{4/3}\right] = \infty

So I got that this integral diverges, however, the book answers are:


Book Answers:

1. ln (2 + \sqrt{3})
2. 0

 

[clustro]

For part 1 at least, the book is correct.

http://en.wikipedia.org/wiki/List_of_integrals_of_irrational_functions

Look at the part "integrals involving s =sqrt(x^2 - a^2) for x^2 > a^2.

I got ln(2 + sqrt(3)) using the formula there.

As for how they got it, I have no idea.

 

[clairez93]

Hmm, I see. I would like to know how this formula was derived though. Also do you see something wrong with #2?

 

[Dick]

For the first one you do a trig substitution, like x=2*sec(t). For the second one the "improper" part of your integrand is it's divergence at x=1. You should be letting x->1 in both halves of the integral, not x->infinity.

 

[g_edgar]

In the first one, \lim_{N \to 0} not \infty

 

[clairez93]

But I set t = x - 1 specifially to make the discontinuity at 0; with t the problem occurs at 0, then don't we take the limit to infinity?

 

[Dick]

Sure, in terms of t the integrand is (-t)^(-1/3) so the discontinuity is at 0. So for example integrate it from epsilon to 1 and let epsilon approach 0. Why would you send anything to infinity? BTW the antiderivative of (-t)^(-1/3) isn't (-3/4)*(-t)^(4/3). What's the correct power?

 

[clairez93]

ah I just flipped through my textbook and you are right. I took the limit to 0 and got the right answer. the first I still am not sure how to integrate, there doesn't seem to be a trig substitution that works

 

[clairez93]

the power you asked for is 2/3

 

[Dick]

ah I just flipped through my textbook and you are right. I took the limit to 0 and got the right answer. the first I still am not sure how to integrate, there doesn't seem to be a trig substitution that works

Your variable changes aren't really helping you. You are just moving the discontinuity around. To do the first integral stick with the form 1/sqrt(x^2-4) and substitute x=2*sec(t).

 

[Dick]

the power you asked for is 2/3

That's better.

 

[clairez93]

I don't understand how you came up with 2 sec t

 

[Dick]

I don't understand how you came up with 2 sec t

If x=2*sec(t) then x^2-4 becomes 4*sec(t)^2-4=4(sec(t)^2-1). sec(t)^2-1=tan(t)^2. Now the quantity inside the square root is a perfect square.

 

[clairez93]

I tried doing the substitution but then got stuck again. This is what I did:

x = 2 sec (t)
dt = 2 sec(t) tan (t) dx
t = sec^{-1} (x/2)

\int^{\pi/3}_{0}\frac{1}{\sqrt{(2 sec(t)^{2} - 4} - 4}(2 sec(t) tan (t)}dt

= \int^{\pi/3}_{0}\frac{1}{\sqrt{4(sec^{2}t -1)}(2 sec(t) tan (t)}dt

= \int^{\pi/3}_{0}\frac{1}{\sqrt{4 tan^{2})t)}(2 sec(t) tan (t)}dt


= \int^{\pi/3}_{0}\frac{1}{2 tan(t)(2 sec(t) tan (t)}dt

= \int^{\pi/3}_{0}\frac{1}{2 tan^{2}(t)(2 sec(t))}dt


However, I still am not sure how to integrate this expression.

 

[Dick]

If x=2*sec(t) then dx=2*sec(t)*tan(t)*dt. NOT dt=2*sec(t)*tan(t)*dx. The tan's will cancel and you should be left with just the integral of sec(t).

 

[Дьявол]

Obviously you did something wrong.
x=2sec(t)

dx=2tan(t)sec(t)dt

Now,
\int^{\pi/3}_{0}\frac{1}{\sqrt{\frac{4}{cos^2(t)}-4}}*2tan(t)sec(t)dt=
=\int^{\pi/3}_{0}\frac{cos(t)}{\sqrt{4(1-cos^2(t))}}*2tan(t)sec(t)dt=
=\int^{\pi/3}_{0}\frac{cos(t)}{2sin(t)}*2tan(t)sec(t)dt=

Do you know how to continue from now on?

You got here cotan(t)*tan(t)=1 and its pretty easy to solve it.

Regards.

 

[clairez93]

Finally solved it! Thank you! I would have never thought to use a trig substitution like that. Thanks.