Prove that two definite integrals are equal

[Char. Limit]

Homework Statement

I want to prove the following statement:

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

Homework Equations

The Attempt at a Solution

To be honest, I'm not sure how to do this. I tried a substitution $$t=e^t$$ 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?

 

[rl.bhat]

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.

 

[╔(σ_σ)╝]

Factor out $$e^{2t}$$ of the first integral and make t= e^t and change the bounds of integration.

 

[Susanne217]

Homework Statement

I want to prove the following statement:

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

Homework Equations

The Attempt at a Solution

To be honest, I'm not sure how to do this. I tried a substitution $$t=e^t$$ 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?

 

[Mark44]

Factor out $$e^{2t}$$ 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.

 

[Mark44]

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)╝.

 

[Susanne217]

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.

 

[╔(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.

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.

 

[Susanne217]

A better approach is to follow the advice given by rl.bhat and ╔(σ_σ)╝.

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 ? ;)

 

[╔(σ_σ)╝]

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.:-)

 

[Susanne217]

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...

 

[╔(σ_σ)╝]

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.

 

[Susanne217]

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...

 

[╔(σ_σ)╝]

Do it and send it as a private message to me :-).
Perhaps you have some clever tricks that are worth seeing.

 

[Susanne217]

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.

 

[╔(σ_σ)╝]

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.

 

[╔(σ_σ)╝]

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.:-)

 

[Susanne217]

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..