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# How to evaluate definite integral if the limits are 0 to infinity? Watch

1. The indefinite integral is , but what is the value of .

On one of the example sheets, ; but how?
2. (Original post by Alpha-Omega)

The indefinite integral is , but what is the value of .
There shouldn't be a square there - you've made a slip in the integration or the partial fractions.

On one of the example sheets, ; but how?
Well the C will cancel, and the lower limit is otherwise 0.

For the upper limit, take the limit as x goes to infinity. It would help to rewrite (1+2x)/(1+x) as 2 - 1/(1+x), in which case the limit should be clear.
3. (Original post by ghostwalker)
There shouldn't be a square there - you've made a slip in the integration or the partial fractions.

Well the C will cancel, and the lower limit is otherwise 0.

For the upper limit, take the limit as x goes to infinity. It would help to rewrite (1+2x)/(1+x) as 2 - 1/(1+x), in which case the limit should be clear.
That C - must be the coffee.

I was wondering about that too, the answer sheet does not have square too, as you say.

I get the following for the partial fraction:

So, shouldn't the integral be:

, and

,

so it must be ?
4. Apart from that, I understand now how ln(2) is achieved.
5. (Original post by Alpha-Omega)
That C - must be the coffee.

I was wondering about that too, the answer sheet does not have square too, as you say.

I get the following for the partial fraction:

So, shouldn't the integral be:

, and

,

so it must be ?
You don't need the 2 in front - remember that d/dx(1+2x) = 2.

To evaluate the limits, decompose it into 1/(1+x) + 2x/(1+x).

As x tends to infinity, 1/(1+x) clearly tends to 0. Now, what happens to 2x/(1+x) as x tends to infinity? Think of how significant (1+x) is compared to x as x gets big (say, 1,000,001 compared to 1,000,000).

As x tends to 0, 1/(1+x) clearly tends to 1, and 2x/(1+x) also has an obvious limit.

What happens when you combine all of these things?
6. (Original post by Hedgeman49)
You don't need the 2 in front - remember that d/dx(1+2x) = 2.

To evaluate the limits, decompose it into 1/(1+x) + 2x/(1+x).

As x tends to infinity, 1/(1+x) clearly tends to 0. Now, what happens to 2x/(1+x) as x tends to infinity? Think of how significant (1+x) is compared to x as x gets big (say, 1,000,001 compared to 1,000,000).

As x tends to 0, 1/(1+x) clearly tends to 1, and 2x/(1+x) also has an obvious limit.

What happens when you combine all of these things?
Ok, I get it now.

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Updated: May 13, 2013
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