Understanding an Integral: Solving and Breaking it Down

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In summary: I have found a great guide online.It is called "Teach Yourself LaTeX in 21 Days" and it is a great guide.I recommend highly that you buy a copy of "Schaum's Outline of Calculus" (Google 'shaums outline series'). It starts at the elementary level and takes you through all the types of things you are asking about here; it has more than 1000 solved problems where you are taken step-by-step through the solution, and it is accompanied by instructional on-line videos. It is not expensive, ranging from about $6 to $14 (used) on Amazon.
  • #1
scottshannon
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Homework Statement


I have posted what appears to be the solution of this integral. I don't understand how the first integral was split apart in the 3rd line.

Homework Equations


The solution is included

The Attempt at a Solution


I have tried to break the integral apart but can't see how to do it...
 

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  • #3
what would be the assignment of u and dv? Does the way that he solved this problem make sense to you?
 
  • #4
Did you understand the second line? The third line is Int(uv)dx= u*int(v)dx- int{u'*(int v dx)}dx

Here int=integration. latex is not working here
 
  • #5
It's a common formula, you must have learned it. If you don't you will find it in your textbook. It's a necessary formula to do integration
 
  • #6
Good day. As Dr Courtney said it is in fact by parts. Once the integral is split into two let:

[tex]u=x[/tex]
[tex]du=dx[/tex]

[tex]v=-2(x^{4}+1)^{-1/2}[/tex]
[tex]dv=(4x^{3})/(x^{4}+1)^{3/2}[/tex]

On the first integral.

Good grief it's nearly impossible to type out latex on an iPad!
 
  • #7
scottshannon said:

Homework Statement


I have posted what appears to be the solution of this integral. I don't understand how the first integral was split apart in the 3rd line.

Homework Equations


The solution is included

The Attempt at a Solution


I have tried to break the integral apart but can't see how to do it...

Here try this - take x2 out from the denominator , and then multiply - divide by two .

See what happens .
 
  • #8
Thank you for your help.

DeldotB..Thank you... I understood that it was probably integration by parts but it wasn't immediately obvious how to make the u and dv assignment.

Qwertywerty..thank you. I took out an x^2 from the denominator and multiplied by 2 on top and bottom and then was able to let u = x^2 +x^(-2) but I would not have been able to have the presence of mind to have taken out the x^2 because I couldn't have foreseen what was going to happen...but thank you.I cannot use Latex and would like to learn...is there a way to learn to use latex?
 
  • #9
scottshannon said:
Thank you for your help.

DeldotB..Thank you... I understood that it was probably integration by parts but it wasn't immediately obvious how to make the u and dv assignment.

Qwertywerty..thank you. I took out an x^2 from the denominator and multiplied by 2 on top and bottom and then was able to let u = x^2 +x^(-2) but I would not have been able to have the presence of mind to have taken out the x^2 because I couldn't have foreseen what was going to happen...but thank you.

What you need is practice .
 
  • #10
I have been trying to find problems of a nature that will allow me to do that...
 
  • #11
scottshannon said:
I have been trying to find problems of a nature that will allow me to do that...

Why don't you ask your teacher / a mentor here ?
 
  • #12
I don't have a teacher or mentor here...I don't know anyone here...
 
  • #13
scottshannon said:
I don't have a teacher or mentor here...I don't know anyone here...

I meant a teacher at your school . You can always post at the academic guidance area if you don't want to ask your teacher .
 
  • #14
Thank you but I don't go to school...I am out of school...I am just trying to learn on my own...
 
  • #15
scottshannon said:
Thank you but I don't go to school...I am out of school...I am just trying to learn on my own...

My bad . I'm not sure how that works .
 
  • #16
Qwertywerty...what was it that led you to take out an from the bottom
 
  • #17
Qwertywerty...what was it that led you to take out an x2 from the bottom
 
  • #18
scottshannon said:
Thank you but I don't go to school...I am out of school...I am just trying to learn on my own...

What resources are you attempting to use? I recommend highly that you buy a copy of "Schaum's Outline of Calculus" (Google 'shaums outline series'). It starts at the elementary level and takes you through all the types of things you are asking about here; it has more than 1000 solved problems where you are taken step-by-step through the solution, and it is accompanied by instructional on-line videos. It is not expensive, ranging from about $6 to $14 (used) on Amazon.

My apologies if you are using this already and it is still not enough.
 
  • #19
scottshannon said:
Qwertywerty...what was it that led you to take out an x2 from the bottom

Well , I could see that taking x2 from the denominator would give you the derivative of the denominator - Practice

I'm sorry if I've said something obvious .
 
Last edited:
  • #20

Related to Understanding an Integral: Solving and Breaking it Down

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve. It is the reverse of differentiation and is used to find the original function when its derivative is known. It is denoted by the symbol ∫ and is an important tool in calculus.

2. How do I solve an integral?

To solve an integral, you need to use integration techniques such as substitution, integration by parts, or trigonometric substitution. First, you need to identify the type of integral you have and then apply the appropriate integration technique. This will help you simplify the integral and make it easier to solve.

3. What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration, while an indefinite integral does not have any limits. A definite integral gives a specific numerical value, whereas an indefinite integral gives a general solution. In other words, a definite integral represents the area under a curve between two points, while an indefinite integral represents a family of curves.

4. How can I break down a complex integral into smaller parts?

To break down a complex integral, you can use integration techniques such as integration by parts or substitution. You can also use the properties of integrals, such as linearity and the power rule, to simplify the integral. Additionally, you can break down the integral into smaller intervals and use the fundamental theorem of calculus to solve each interval separately.

5. Why is it important to understand integrals?

Integrals are an essential tool in calculus and are used in various fields such as physics, engineering, and economics. They help us find areas, volumes, and other important quantities that cannot be easily calculated by other methods. Understanding integrals also helps us better understand the concept of change and the relationship between a function and its derivative.

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