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Changing limits of integration to infinity watch

1. I am doing the final part of the question, so using the substitution suggested:

when

when

I don't understand how you can say this (*), surely it is undefined?
2. (Original post by lizard54142)

I am doing the final part of the question, so using the substitution suggested:

when

when

I don't understand how you can say this (*), surely it is undefined?
It's a very minor abuse of notation. If you restrict theta to the interval 0 to pi/2 then as theta approaches pi/2, x increases unboundedly i.e. it approaches positive infinity. We just write the infinity symbol as the upper limit even though it's not actually a number to represent what's really going on.
3. (Original post by davros)
It's a very minor abuse of notation. If you restrict theta to the interval 0 to pi/2 then as theta approaches pi/2, x increases unboundedly i.e. it approaches positive infinity. We just write the infinity symbol as the upper limit even though it's not actually a number to represent what's really going on.

Okay, yeah that's what I originally thought. Say then we define an integral as:

let

Using this substitution, how would the limits change in this case?
4. (Original post by lizard54142)

Okay, yeah that's what I originally thought. Say then we define an integral as:

let

Using this substitution, how would the limits change in this case?
You need to be careful because 1/x is again undefined at 0. If a > 0 then you're using that branch of 1/x which approaches +infinity as x->0 from above so again you will use +infinity as the limit of integration.
5. (Original post by davros)
You need to be careful because 1/x is again undefined at 0. If a > 0 then you're using that branch of 1/x which approaches +infinity as x->0 from above so again you will use +infinity as the limit of integration.
Okay, so the new integral would be:

Is this correct?

Also another question, say we have evaluated a definite integral and have this:

Is this equal to:

6. (Original post by lizard54142)
Okay, so the new integral would be:

Is this correct?

Also another question, say we have evaluated a definite integral and have this:

Is this equal to:

You need a 1/u^2 in your integral - you can't leave it as a mixture of u's and x's!

And yes, your final expression is how we define how to evaluate a definite integral when one of the limits is infinite.
7. (Original post by davros)
You need a 1/u^2 in your integral - you can't leave it as a mixture of u's and x's!

And yes, your final expression is how we define how to evaluate a definite integral when one of the limits is infinite.
Yeah I understand you can't have a mixture, I just left it like that because I was lazy okay I think I have this sorted now, thanks.

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