Explaining Basic Integration: dv/v, Natural Log of V2/V1

In summary, the conversation discusses the integration of dv/v between V2 and V1 and the resulting value of natural log of V2/V1. The use of absolute value signs for v1 and v2 is also discussed, as well as the confusion and clarification around integrating from v2 to v1 versus v1 to v2. The final consensus is that integrating from v2 to v1 gives ln|v2| - ln|v1| = ln|v2/v1|.
  • #1
jamesd2008
64
0
Hi

could someone explain to me why, if I integrate dv/v between V2 and V1 the result is nastural log of V2/V1?

Thanks
 
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  • #2
Is your question why the primitive of 1/v is log v or are you wondering why you get log v2/v1 instead of log v2 - log v1? Anyhow [itex]\log v_2-\log v_1=\log v_2/v_1[/itex].
 
  • #3
Yes thanks Cyosis!

Missed the fact that logV2-logV1=LogV2/V1

Thank you very much for your insight!
james
 
  • #4
Also is not that 1/v.dv between v2 and v1 = [log|v2|-log|v1|] between v2 and v1? and that the dv is ignored?
 
  • #5
The dv is not ignored, it is necessary in an integral to specify the variable of integration.
 
  • #6
jamesd2008 said:
Also is not that 1/v.dv between v2 and v1 = [log|v2|-log|v1|] between v2 and v1? and that the dv is ignored?

can you restate this more clearly? I don't get what you are asking.
 
  • #7
jamesd2008 said:
could someone explain to me why, if I integrate dv/v between V2 and V1 the result is nastural log of V2/V1?
[tex]\text{By the above phrase I think most people would understand }\int_{v_2}^{v_1}\frac{dv}{v}=log_e(v_1/v_2)\text{ not }log_e(v_2/v_1)\text{.}[/tex]
jamesd2008 said:
Also is not that 1/v.dv between v2 and v1 = [log|v2|-log|v1|] between v2 and v1? and that the dv is ignored?
[tex]\text{Does the use of }|v_1|,|v_2|\text{ here mean that you're interested in values of }v_1\text{ and }v_2\text{ other than positive and real?}[/tex]
 
Last edited:
  • #8
Hi, I think what I'm getting confused about, is that for entropy the change in entropy is ds=dQ/T. Integrating this gives s2-s1=the integral of dQ/T between 2 and 1. So are you saying that the change in Q is now just there to specify the variable of integration? Sorry id this all sounds confusing.
 
  • #9
It does, apology accepted :)
 
  • #10
integral (dv / v) = integral (1/v)*dv = integral (1/v) dv = ln|v|

since our integral is between V2 and V1, we do ln|V2| - ln|V1| = ln|V2/V1|
 
  • #11
luma said:
...
since our integral is between V2 and V1, we do ln|V2| - ln|V1| = ln|V2/V1|
Martin Rattigan said:
[tex]\text{By the above phrase I think most people would understand }\int_{v_2}^{v_1}\frac{dv}{v}=log_e(v_1/v_2)\text{ not }log_e(v_2/v_1)\text{. ...}[/tex]
Martin Rattigan said:
[tex]\text{ ... Does the use of }|v_1|,|v_2|\text{ here mean that you're interested in values of }v_1\text{ and }v_2\text{ other than positive and real?}[/tex]

(As before.)
 
  • #12
Thanks everyone for there help.
 
  • #13
Martin Rattigan said:
[tex]\text{By the above phrase I think most people would understand }\int_{v_2}^{v_1}\frac{dv}{v}=log_e(v_1/v_2)\text{ not }log_e(v_2/v_1)\text{.}[/tex]

[tex]\text{Does the use of }|v_1|,|v_2|\text{ here mean that you're interested in values of }v_1\text{ and }v_2\text{ other than positive and real?}[/tex]

If you integrate from v2 to v1, yes, that is correct. I was under the impression that e was integrating from v1 to v2. The word "between" creates an ambiguity there, I suppose.
 

1. What is integration and why is it important?

Integration is a mathematical process of finding the area under a curve. It is important because it allows us to solve a wide range of problems in various fields, such as physics, engineering, and economics.

2. What is dv/v in basic integration?

In basic integration, dv/v refers to a method used to integrate functions of the form 1/v. It involves substituting u = ln(v) and dv = 1/v, which simplifies the integration process.

3. How is the natural log of V2/V1 used in integration?

The natural log of V2/V1 is used to solve integrals involving exponential functions, as it allows us to rewrite the function as ln(V2) - ln(V1). This makes it easier to integrate using the basic integration rules.

4. What are some applications of integration in real life?

Integration is widely used in real life, including calculating areas and volumes in construction and architecture, determining displacement and velocity in physics, and finding optimal solutions in economics and business.

5. Is there a shortcut to solving integrals?

While there are some techniques, such as substitution and integration by parts, that can make integration easier, there is no universal shortcut to solving integrals. It requires practice and understanding of the basic integration rules and techniques.

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