Need help on a simple but frustrating integral

  • Thread starter Deuterium2H
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In summary: That makes much more sense.In summary, while I understand much of the complex analysis, I sometimes get hung up over the what I believe are rather elementary integrals which are never solved in a step by step fashion...but are just shown and then (with no explanation) given a solution. This can be a bit frustrating, especially if it has been years (decades) since one had advanced Calc, and has forgotten many of the common integral forms/solutions.
  • #1
Deuterium2H
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I am reading Paul J. Nahn's excellent "Mrs. Perkins's Electric Quilt: and other Intriquing Stories of Mathematical Physics".

Unfortunately, while I can understand much of the complex analysis, I sometimes get hung up over the what I believe are rather elementary integrals which are never solved in a step by step fashion...but are just shown and then (with no explanation) given a solution. This can be a bit frustrating, especially if it has been years (decades) since one had advanced Calc, and has forgotten many of the common integral forms/solutions.

Anyways, I didn't even get through the damn Preface without encountering a problem.

We are given the following differential equation (integral):

ds = [ v0 / (1 + kv0t) ] dt

k is a Constant of proportionality, and v0 is initial velocity, which obviously itself is a function of time.

The solution is given as:

s = ln[(1 + kv0t)(1/k)] + Z

How does one integrate the right side of the equation? I can get the sense that the solution will be a natural logarithm, due to the fact that we have a form INTEGRAL (dt/t).
Would someone be so kind as to break this down step by step for me. I am almost embarrassed to ask, but oh well. Thanks in advance!
 
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  • #2
Ah sugar...I think I might have remembered something. Is this one of those instances where we have an integral in which the numerator is a derivative of the denominator, and evaluated by the formula:

INT [ f'(x) / f(x) ] dx = ln |f(x)| + C ??

So,

s = (1/k) INT [ (kv0)/(1 + kv0t) ] dt
 
Last edited:
  • #3
On further thought, I guess that does solve it. Would appreciate someone else's confirmation, however. Thanks again
 
  • #4
Yup. That's just about right.
 
  • #5
Char. Limit said:
Yup. That's just about right.

Thanks Char. Limit,

The only thing troubling me (still) is the fact that v0 is itself a funtion of "t"...and it almost seems like there is a partial derivative involved. However, if I just treat
d(kv0t)/dt = kv0, then it all works out simply.
 
  • #6
Are you sure that v0 is a function of t?

EDIT: Initial velocity is actually NOT a function of time. Indeed, it's not a function at all! It's a constant value.
 
  • #7
Char. Limit said:
Are you sure that v0 is a function of t?

EDIT: Initial velocity is actually NOT a function of time. Indeed, it's not a function at all! It's a constant value.

AHHHHH Yes!

Once again, thanks Char. Limit. You are absolutely right, and I had totally overlooked that.
 

Related to Need help on a simple but frustrating integral

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is used to calculate the total value of a function over a certain interval.

2. Why is this integral frustrating?

This integral may be frustrating because it is not easily solvable using basic integration techniques and may require more advanced methods or the use of a computer program to solve.

3. How can I solve this integral?

There are several methods for solving integrals, including basic integration techniques such as substitution and integration by parts, as well as more advanced techniques like partial fractions or using a computer program. It is important to understand the properties and rules of integration before attempting to solve a specific integral.

4. What are the applications of integrals?

Integrals have many applications in mathematics, physics, and engineering. They are used to calculate areas, volumes, and other physical quantities, as well as in optimization problems and differential equations.

5. How can I improve my integration skills?

Practice is key to improving integration skills. It is important to have a solid understanding of basic integration techniques and to work through a variety of problems. Seeking help from a tutor or using online resources can also be helpful in improving integration skills.

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