Prove that two definite integrals are equal

In summary: If you mean your non linear differential equation then, no. I do not know how to solve it. I haven't taking any courses on non linear differential equations neither am I interested in learning how to do so,...
  • #1
Char. Limit
Gold Member
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Homework Statement


I want to prove the following statement:

[tex]\int_0^{ln(2)} \sqrt{e^{2t} + 4 e^{4t} + 9 e^{6t}} dt = \int_1^2 \sqrt{1 + 4 t^2 + 9 t^4} dt[/tex]


Homework Equations





The Attempt at a Solution



To be honest, I'm not sure how to do this. I tried a substitution [tex]t=e^t[/tex] for the second integral, but it didn't really go all that well. How do you do this, assuming that direct calculation is not allowed?
 
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  • #2
Let e^t = x.
Then e^t*dt = dx,

When t = 0, x = 1 and when t = ln(2), x = 2.
Substitute these values in the given integration.
Then replace x by t, because the definite integration does not depend on the variables whether it is x or t or any other.
 
  • #3
Factor out [tex]e^{2t}[/tex] of the first integral and make t= e^t and change the bounds of integration.
 
  • #4
Char. Limit said:

Homework Statement


I want to prove the following statement:

[tex]\int_0^{ln(2)} \sqrt{e^{2t} + 4 e^{4t} + 9 e^{6t}} dt = \int_1^2 \sqrt{1 + 4 t^2 + 9 t^4} dt[/tex]


Homework Equations





The Attempt at a Solution



To be honest, I'm not sure how to do this. I tried a substitution [tex]t=e^t[/tex] for the second integral, but it didn't really go all that well. How do you do this, assuming that direct calculation is not allowed?

I seem to remember if you have two Riemann sums R1 and R2 if they converge to the same point then they are equal. Could this be used here?
 
Last edited:
  • #5
╔(σ_σ)╝ said:
Factor out [tex]e^{2t}[/tex] of the first integral and make t= e^t and change the bounds of integration.
It's less confusing to use a different variable; say, u = e^t.
 
  • #6
Susanne217 said:
I seem to remember if you have two Riemann sums R1 and R2 if they converge to the same point then they are equal. Could this be used here?
A better approach is to follow the advice given by rl.bhat and ╔(σ_σ)╝.
 
  • #7
Mark44 said:
A better approach is to follow the advice given by rl.bhat and ╔(σ_σ)╝.

Why is that? Cause if the Riemann sums Converge to the same number then integrals are equal aren't they? Maybe I'm just a silly girl.
 
  • #8
Susanne217 said:
Why is that? Cause if the Riemann sums Converge to the same number then integrals are equal aren't they? Maybe I'm just a silly girl.
First of all you have to write out the riemann sums and show it converges to something and write out down the something.

Then you have to prove the two riemann sums are equal.
 
  • #9
Mark44 said:
A better approach is to follow the advice given by rl.bhat and ╔(σ_σ)╝.

╔(σ_σ)╝ said:
First of all you have to write out the riemann sums and show it converges to something and write out down the something.

Then you have to prove the two riemann sums are equal.

Well then I wasn't far of was I ? ;)
 
  • #10
Susanne217 said:
Well then I wasn't far of was I ? ;)
Well your suggestion was not "wrong". However, it will not be helpful to the OP as the technicality involving your suggestion require a lot of work. And remember the riemann sum ,by definition, is not something you can easily manipulate. When you have functions that are not polynomials , integrating from the definition becomes non trivial even for relatively "simple" functions.

How about you try out your suggestion and see if it works out well.:-)
 
  • #11
╔(σ_σ)╝ said:
Well your suggestion was not "wrong". However, it will not be helpful to the OP as the technicality involving your suggestion require a lot of work. And remember the riemann sum ,by definition, is not something you can easily manipulate. When you have functions that are not polynomials , integrating from the definition becomes non trivial even for relatively "simple" functions.

How about you try out your suggestion and see if it works out well.:-)

If I do I will properly get an infraction...
 
  • #12
You don't have to post it here and even if you do, you would not get an infraction. Just confess if you cannot do it.
 
  • #13
╔(σ_σ)╝ said:
You don't have to post it here and even if you do, you would not get an infraction. Just confess if you cannot do it.

I posted something simular last year and got an infraction so. But yes I could do it...
 
  • #14
Do it and send it as a private message to me :-).
Perhaps you have some clever tricks that are worth seeing.
 
  • #15
╔(σ_σ)╝ said:
Do it and send it as a private message to me :-).
Perhaps you have some clever tricks that are worth seeing.

If you can solve the differential in my other post, then we have a deal.
 
  • #16
Susanne217 said:
If you can solve the differential in my other post, then we have a deal.
If you mean your non linear differential equation then, no. I do not know how to solve it. I haven't taking any courses on non linear differential equations neither am I interested in learning how to do so, atm.
 
  • #17
Susanne217 said:
If you can solve the differential in my other post, then we have a deal.
If you mean your non linear differential equation then, no. I do not know how to solve it. I haven't taking any courses on non linear differential equations neither am I interested in learning how to do so, atm.
I know my limitations.:-)
 
  • #18
╔(σ_σ)╝ said:
If you mean your non linear differential equation then, no. I do not know how to solve it. I haven't taking any courses on non linear differential equations neither am I interested in learning how to do so, atm.

yeah it was this one here

https://www.physicsforums.com/showthread.php?t=431007

OKay fair enough..
 

Related to Prove that two definite integrals are equal

1. What is the definition of a definite integral?

A definite integral is a mathematical concept used to calculate the area under a curve between two specific points on a graph. It is represented by the symbol ∫ and is often used to find the total value of a function over a certain interval.

2. How do you prove that two definite integrals are equal?

To prove that two definite integrals are equal, you must show that they have the same value for any given interval. This can be done by using the properties of integrals, such as linearity and the fundamental theorem of calculus, and by evaluating both integrals using techniques such as substitution or integration by parts.

3. Can two definite integrals be equal even if the functions being integrated are different?

Yes, two definite integrals can be equal even if the functions being integrated are different. This is because the value of a definite integral is determined by the area under the curve, not the specific function itself. As long as the area under both curves is the same, the integrals will be equal.

4. What are some common techniques used to prove the equality of two definite integrals?

Some common techniques used to prove the equality of two definite integrals include substitution, integration by parts, and the properties of integrals such as linearity and the fundamental theorem of calculus. Other methods may also be used depending on the specific functions being integrated.

5. Why is it important to prove that two definite integrals are equal?

Proving the equality of two definite integrals is important because it ensures that the area under the curves of the two functions is the same. This can be useful in many applications, such as calculating volumes and areas in physics and engineering or finding probabilities in statistics. It also helps to establish the validity and accuracy of mathematical equations and models.

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