Where to Begin with Integrals to Integration by Parts

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In summary, the conversation discusses techniques for solving integrals, specifically the Cauchy formula for iterated integrals and the use of integration by parts. The importance of not expanding the integral and instead using the substitution (x-t)^2 = g(t) is also emphasized.
  • #1
russia123
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I've looking at this and I'm dumbfound as to where to begin. Integrals have never been my strong suit.

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  • #2
This is called the Cauchy formula for iterated integrals (don't mix it up with the Cauchy formula in complex analysis). Ignore the left hand side. Suppose you were asked to answer the right hand side in an exam. What techniques do you know which would help you?
 
  • #3
What I had in mind is expanding the (x-t)^2, and then multiplying everything out, and then I would have 3 separate integrals due to being able to separate integrals based on addition.
 
  • #4
No, don't expand. If I wrote (x-t)2 = g(t), would that give you ideas?
 
  • #5
Ah, integration by parts is the first thing that comes to mind. Don't know how I missed that.
 

Related to Where to Begin with Integrals to Integration by Parts

1. How do I prove an integral?

The process of proving an integral involves using mathematical techniques such as substitution, integration by parts, or trigonometric identities to manipulate the integral into a form that is easier to solve. It is important to understand the fundamental principles of integration and have a strong grasp of algebra before attempting to prove an integral.

2. What are the steps for proving an integral?

The steps for proving an integral may vary depending on the specific integral being solved, but generally, the process involves identifying the appropriate method of integration, manipulating the integral to a solvable form, and then solving it using integration rules or techniques. It may also involve evaluating the integral at certain limits or using additional mathematical properties to simplify the solution.

3. Can I use a calculator to prove an integral?

While a calculator may be helpful in checking your work, it is not recommended to solely rely on a calculator to prove an integral. Understanding the concepts and techniques behind integration is crucial in solving integrals and relying too heavily on a calculator may hinder your understanding and ability to solve more complex integrals.

4. How do I know if my integral is correct?

To check the correctness of your integral, you can differentiate the solution and see if it yields the original function. This is known as the fundamental theorem of calculus. Additionally, you can also check your solution using a calculator or by comparing it to known solutions or tables of integrals.

5. Are there any tips for proving integrals more efficiently?

Some tips for proving integrals more efficiently include practicing frequently, understanding the properties and rules of integration, and being familiar with common techniques such as substitution and integration by parts. It is also helpful to break down the integral into smaller parts and to check your work along the way to avoid making mistakes.

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