Integrating with Partial Fractions and Completing the Square: Tips and Tricks"

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In summary, The conversation is about solving an integral using partial fractions and trigonometric substitution. The expert suggests expressing the numerator as (2x-2) + 3 and splitting the expression into two integrals. They mention two ways to solve the second integral - using a hyperbolic trig sub or factoring and using partial fractions. The expert also explains that they recognized the form f'(x)/f(x) in the integrand. The original speaker asks for further explanation and the expert explains that they figured out the necessary term to add to the derivative to make it the numerator in the integrand.
  • #1
Dan350
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I tried Partial fractions,, I completed the square which is (x-1)^2-3 ,, but,, what to do with the numer 3? i tried trig sub also,, didnt worked,,
Any ideas on how to integrate?
Is there a way to simplify??

Thanks a lot,,
 

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  • #2
Dan350 said:
I tried Partial fractions,, I completed the square which is (x-1)^2-3 ,, but,, what to do with the numer 3? i tried trig sub also,, didnt worked,,
Any ideas on how to integrate?
Is there a way to simplify??

Thanks a lot,,

Express the numerator as (2x-2) + 3.

Split up the expression. The first integrand is of the form f'(x)/f(x).

For the second integral, it's probably easiest to just factorise the expression, then use partial fractions. You may have some surds to deal with, but it's still quite simple.

Sorry for the multiple edits, I'm rushing for time and am doing this in my head.
 
  • #3
Curious3141 said:
Express the numerator as (2x-2) + 3.

Split up the expression. The first integrand is of the form f'(x)/f(x).

Second integral can be done in one of two ways.

First is to complete the square as you've done, then use a hyperbolic trig sub (think cosh).

Second is to factorise the expression, then use partial fractions.


,, but why did you express it like that? (2x-2) + 3 thanks a lot, is see I can use use u sub since d/dx of x^2-2x-2 = 2x-2.. but i woul like to know how and why did you expressit like that,... thank you!
 
  • #4
Dan350 said:
,, but why did you express it like that? (2x-2) + 3 thanks a lot, is see I can use use u sub since d/dx of x^2-2x-2 = 2x-2.. but i woul like to know how and why did you expressit like that,... thank you!

I hope you took notice of my edited post.

Why did I express it like that? It's just pattern recognition. I guess I just taught myself to recognise that form f'(x)/f(x) "hiding" in an integrand.
 
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  • #5
Curious3141 said:
I hope you took notice of my edited post.

Why did I express it like that? It's just pattern recognition. I guess I just taught myself to recognise that form f'(x)/f(x) "hiding" in an integrand.

? but the 3 , where did you get it from? Would you mind to explain me really quick? Thanks
 
  • #6
(2x-2)+3 = ?
 
  • #7
SteamKing said:
(2x-2)+3 = ?

Yess
The procedure,, i would really appreciate
 
  • #8
Dan350 said:
Yess
The procedure,, i would really appreciate

I figured out what f'(x) would be, then what I had to add on to it to make the numerator in the integrand.
 

FAQ: Integrating with Partial Fractions and Completing the Square: Tips and Tricks"

1. What is an integral?

An integral is a mathematical concept used to find the area under a curve in a graph. It is also used to calculate displacement, volume, and other quantities in physics and engineering. In simpler terms, an integral is a way to find the total value of something that is continuously changing.

2. Why do I need help with an integral?

Integrals can be complex and difficult to solve, especially for those who are not familiar with advanced mathematical concepts. Seeking help with an integral can provide a better understanding of the problem and lead to a correct solution.

3. How do I solve an integral?

To solve an integral, you can use various techniques such as substitution, integration by parts, or partial fractions. It is essential to understand the fundamental principles of integration and practice solving different types of integrals to become proficient in solving them.

4. Can I use a calculator to solve an integral?

Yes, there are calculators and software programs that can solve integrals. However, it is important to have a basic understanding of the concepts behind integrals and double-check the results provided by the calculator to ensure accuracy.

5. Are integrals used in real-life applications?

Yes, integrals are widely used in various fields, including physics, engineering, economics, and statistics. They are used to calculate quantities such as displacement, work, and probability, making them essential in understanding and solving real-world problems.

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