Understanding Integration by Substitution

In summary, the conversation is about a solution to an integration problem using a different method than the conventional one. The first line shows the problem and the user asks for an explanation of the second line. The response clarifies that it is a different way of writing integration by substitution and the second line is a step in the integration process. The user then shares their attempt at solving the problem using integration by parts and asks for further clarification. The final response explains that the second line is a result of two successive integrations by parts.
  • #1
Delzac
389
0

Homework Statement


[PLAIN]http://img293.imageshack.us/img293/5026/solutoni.png

Hi all,

Can anyone explain what is going on where? I understand that it is a different way of writing the conventional integration by substitution, instead of using the symbol u. The second line, however, especially confuse me. I don't understand how the working changes between d(sin2010x) and dx.



The Attempt at a Solution


I used integration by parts and worked out the answer to be zero.

Double checked with online integration calculator.

Any help would be greatly appreciated.

Delzac
 
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  • #2
Hi Delzac! :smile:
Delzac said:
The second line, however, especially confuse me. I don't understand how the working changes between d(sin2010x) and dx.

As you say, it start by avoiding using a "u".

The jump you don't understand is an integration by parts in which the […] bit has been omitted because it's zero. :rolleyes:

(never do that in an exam!)

The whole thing is just a (to my mind) rather long-winded and confusing way of doing two successive integrations by parts. :redface:
 
  • #3
Oh! now i see. Thanks for the help!

Delzac
 

Related to Understanding Integration by Substitution

1. What is integration by substitution?

Integration by substitution is a method used in calculus to evaluate integrals. It is also known as the u-substitution method. The method involves substituting a variable in the integral with a new variable to simplify the integral and make it easier to solve.

2. When is integration by substitution used?

Integration by substitution is used when an integral involves a composition of functions. This means that the integral has a function within another function. The method is particularly useful when the inner function can be substituted with a new variable to simplify the integral.

3. How do you perform integration by substitution?

To perform integration by substitution, follow these steps:

  • Identify a function within the integral that can be substituted with a new variable.
  • Choose a new variable and find its derivative.
  • Substitute the function and its derivative into the integral, replacing the old variable with the new one.
  • Simplify the integral and solve for the new variable.
  • Replace the new variable with the original function to get the final answer.

4. What are the benefits of using integration by substitution?

Integration by substitution can make solving integrals easier and more efficient. It can also help to solve integrals that may be difficult or impossible to solve using other methods. Additionally, it can be used to solve a wide range of integrals, making it a versatile tool in calculus.

5. Are there any limitations to integration by substitution?

Integration by substitution is not always applicable to every integral. It can only be used when there is a composition of functions in the integral. If the integral does not have this form, then the method cannot be used. Additionally, it may not always lead to a simpler integral, making it necessary to use other integration techniques in some cases.

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