How Can These Two Integral Identities Be Proven?

In summary, the conversation discusses two integral identities and the process of proving them. The first identity is easily proved using integration by parts, but the second one is more difficult and the individual asking for help has been unable to figure it out. They are reminded to show their efforts and not get discouraged in trying to solve the problem.
  • #1
jarvisyang
5
0
Prove two integral identities?

1. The following integral identity holds
[tex]\dfrac{d}{dx}\intop_{x}^{a}\dfrac{F(\rho)d\rho}{\sqrt{\rho^{2}-x^{2}}}=-\dfrac{F(a)x}{a\sqrt{a^{2}-x^{2}}}+x\intop_{x}^{a}\dfrac{d\rho}{\sqrt{\rho^{2}-x^{2}}}\dfrac{d}{d\rho}\left[\dfrac{F(\rho)}{\rho}\right][/tex]
Hints: this can easily proved by applying ingtegration by parts to the right hand side of the identity
2. But the following can also hold
[tex]\dfrac{d}{dx}\intop_{x}^{a}\dfrac{F(\rho)d\rho}{\sqrt{\rho^{2}-x^{2}}}=-\dfrac{F(a)a}{x\sqrt{a^{2}-x^{2}}}+\dfrac{1}{x}\intop_{x}^{a}\dfrac{\rho d\rho}{\sqrt{\rho^{2}-x^{2}}}\dfrac{d}{d\rho}F(\rho)[/tex]
I can not figure out the second identity.Is there anybody can help me?I'm waiting for your excellent proof!
 
Physics news on Phys.org
  • #2
We are not allowed to help in problems unless you demonstrate an effort to solve the problem yourself.
 
  • #3
pwsnafu said:
We are not allowed to help in problems unless you demonstrate an effort to solve the problem yourself.
Actually, I have made a lot efforts. The first identity has been proved by myself. But as for the second identities, I have been thinking for a long time and I still can not figure it out.
 
  • #4
jarvisyang said:
Actually, I have made a lot efforts. The first identity has been proved by myself. But as for the second identities, I have been thinking for a long time and I still can not figure it out.

Well, you know what you do in that case don't you? Show us what you tried even if it's stupid-looking. You know good cooks try again don't you? Yeah, they mess up but they don't get discouraged, then try the recipie again, and eventually they cookin' with kerosene and you wonder how they got so good. Try the recepie even if you burn the dish. The trying part is important.
 
  • #5
jackmell said:
Well, you know what you do in that case don't you? Show us what you tried even if it's stupid-looking. You know good cooks try again don't you? Yeah, they mess up but they don't get discouraged, then try the recipie again, and eventually they cookin' with kerosene and you wonder how they got so good. Try the recepie even if you burn the dish. The trying part is important.

OK.Thank you for your suggestion, jackmell. I've got it. I will post my question together with my efforts or scripts next time.
 

FAQ: How Can These Two Integral Identities Be Proven?

What are integral identities?

Integral identities are mathematical equations that relate to integrals, which are a type of mathematical operation that calculates the area under a curve. Integral identities often involve trigonometric functions and are useful in solving various problems in calculus and other branches of mathematics.

How do I prove two integral identities?

There are several methods for proving two integral identities, depending on the specific identities involved. One common method is to use algebraic manipulation to transform one side of the equation into the other. Another approach is to use substitution or integration by parts to simplify the integrals on both sides of the equation. It is also important to carefully consider the domain of the integrals and any restrictions on the variables.

Can integral identities be used in real-world applications?

Yes, integral identities can be applied to real-world problems in fields such as physics, engineering, and economics. For example, the fundamental theorem of calculus, which is an important integral identity, is used in calculating rates of change and in finding areas and volumes of irregular shapes.

Are there any common mistakes to avoid when proving two integral identities?

One common mistake is to assume that two integrals with the same integrand are equal. This is not always the case, as the limits of integration and other factors can affect the value of the integral. Another mistake is to manipulate both sides of the equation at the same time without keeping track of the changes made, which can lead to incorrect results.

How can I practice proving integral identities?

The best way to practice proving integral identities is to work through problems and exercises in a textbook or online resource. It is also helpful to review the properties of integrals and trigonometric functions and to familiarize yourself with common techniques for solving integrals. Additionally, seeking guidance from a teacher or tutor can provide valuable feedback and help you improve your skills in proving integral identities.

Similar threads

Higher dimensional FTC in electrostatics: Mathematically rigorous or not?
Replies
5
Views
1K
I Multiple integral Jacobian confusion
Replies
7
Views
2K
I Are Extra Conditions Affecting the Limit Definition?
Replies
1
Views
2K
MHB Q1 Can you pass this 3 question AP calculus Quiz in 10 minutes
Replies
3
Views
941
I Understanding a proof of inexistence of max nor min
Replies
4
Views
1K
MHB How can I evaluate the integral using a trigonometric identity?
Replies
5
Views
1K
B Why do I not understand what seems to be basic Calculus?
Replies
11
Views
2K
MHB 4.1.237 AP calculus exam find area
Replies
4
Views
2K
I Is interchanging the order of the surface and volume integrals valid here?
Replies
3
Views
2K
I Why is this closed line integral zero?
Replies
8
Views
2K

Hot Threads

Recent Insights

Back
Top