Integrating Fractions: A Comprehensive Guide

In summary, the speaker is struggling to integrate a fraction expression and asks for help. The other person suggests using a trigonometric substitution and provides a link to a table of integrals. The speaker then attempts to apply the suggested substitution and asks for confirmation. The other person confirms and encourages the speaker to continue working on the problem.
  • #1
Lodve
11
0
I really struggle to integrate this fraction expression:


[tex] \int \frac{2}{4x^2+1}[/tex]
 
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  • #2
I presume you're familiar with the standard table of integrals?

http://www.maths.abdn.ac.uk/~igc/tch/ma1002/int/node51.html

Here's a list of many common ones, see if you can get it into the right form for one of them.

If you have to work this out by hand, consider using a trigonometric substitution, but choose carefully.

(Hint: the relationship 1 + x^2 = sec^2 for x = tan x)
 
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  • #3


Yes, I'm fimiliar with some the of standard table of integrals. I think I just change the equation above in the form
[tex] \int \frac{1}{x^2+a^2}[/tex] where a is an integer, so that the integration of the equation would likely be
[tex] \frac{1}{a} arctan(x/a) [/tex]

But I'm not sure if I've done it right.
 
  • #4


You chose the right one, try to get it into a form like that.
 
  • #5


Thank you for helping me :D
 

1. What is a standard table of integrals?

A standard table of integrals is a reference guide containing a list of commonly used integrals and their corresponding solutions. It is a useful tool for mathematicians, engineers, and scientists who regularly encounter integration problems in their work.

2. How is a standard table of integrals organized?

A standard table of integrals is typically organized into sections based on the types of integrals, such as algebraic, trigonometric, exponential, and logarithmic. Within each section, the integrals are arranged in order of increasing difficulty.

3. How can I use a standard table of integrals?

To use a standard table of integrals, you must first identify the type of integral you are working with. Then, find the corresponding section in the table and locate the specific integral you need. Finally, use the solution provided in the table to solve your integration problem.

4. Are all integrals included in a standard table of integrals?

No, a standard table of integrals typically only includes the most commonly used integrals. It may not have solutions for more complex or specialized integrals. In these cases, other methods, such as numerical integration, may be necessary.

5. Can I rely on a standard table of integrals for all my integration problems?

While a standard table of integrals can be a helpful resource, it is important to understand the principles of integration and be able to derive solutions on your own. Additionally, the table may not have solutions for all integrals, so it is important to have alternative methods for solving integration problems.

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