Solving Improper Integrals: Converging or Diverging?

[Timebomb3750]

Just got into improper integrals, in my Calculus 2 class. We're looking to see if the integral converges or diverges.

Homework Statement

The integral given:
∫(dt/(t+1)^2) on the interval from -1 to 5

Homework Equations

uhhh...

The Attempt at a Solution

Took the limit as "a" goes to -1.

Did a simple u substitution with u=t+1, so that du=dt.

So, you're left with (du/u^2)

The integral of that is -(1/u) meaning -(1/(t+1))

Then I used the fundamental theorem of calculus by evaluating the integral from 5 to "a".

That looks like: -(1/(5+1)) - (1/(a+1))

Basically, I have no clue if I'm doing this right. According to the back of the book, it diverges. But I have no idea how to see that. Any help would be appreciated. Thanks.

 

[dynamicsolo]

You have
$$\int_{-1}^{5} \frac{dt}{(t+1)^{2}} = -\frac{1}{6} - [ \lim_{a \rightarrow -1} (-\frac{1}{a+1}) ] .$$

What happens when you apply the limit?

 

[Timebomb3750]

You mean apply the -1 into "a"? That would make it undefined because you'd be dividing by zero. But what does this tell me about convergence/divergence?

 

[dynamicsolo]

And so that limit is undefined. Therefore, there is no meaningful value for the integral: that is what is meant by "divergence". We say the integral "converges" if the limit for its value approaches a finite number. If the limit for the integral does not approach a finite value, or does not even exist, the integral is said to "diverge". (It should also give the definitions of convergence and divergence of an integral in your textbook.)

 

[Timebomb3750]

Wow. Thanks for clearing that up. I'm starting to understand it now, as I do more problems. Sometimes, I can't understand my textbook, nor my professor.