Help with integrals

[grayson]

Hello, my name is Grayson. I am a bit ahead of myself, learning calculus. I understood like, 70 percent of it but I need help with integrals. My "Calculus for dummies book" Isn't explaining things good enough for me. So can anyone give me a step-by-step guide on how to solve integrals and what the best graphing calculator to use is. Because I don't have a ti-84 and all the online ones work drastically different.

[Genady]

10 minutes ago, grayson said:

a step-by-step guide on how to solve integrals

Such a thing does not exist. However, there are many tools which help. And trial and error.

[grayson]

12 minutes ago, Genady said:

Such a thing does not exist. However, there are many tools which help. And trial and error.

What are some tools that can help? and by a step-by-step guide I mean a less intimidating way to understand integrals. Calculus for dummies was easy. Instead of the complex power rule formula, it gave me a guide. Instead of f=2^3 you put the three before the two and than you add two (which is three minus one). When It got to the Rehmann sum (or however you spell it) I got to intimidated and left the book. That is how my mental disorder brain works (adhd, autism, who knows what)

Edit: I explained the power rule bad. Instead of 2^3 it is 3 2^2

Edited August 19, 2023 by grayson

[Genady]

10 minutes ago, grayson said:

a less intimidating way to understand integrals

What don't you understand?

[grayson]

1 minute ago, Genady said:

What don't you understand?

I don't understand the indefinite integral, And I don't understand what goes into an integral. Sometimes I put integrate f(x)=dx into a calculus calculator and nothing comes out.

 

[Genady]

5 minutes ago, grayson said:

I don't understand the indefinite integral, And I don't understand what goes into an integral. Sometimes I put integrate f(x)=dx into a calculus calculator and nothing comes out.

 

Seems to me that it's better to start from the other end.

What do you know about integrals?

[grayson]

Just now, Genady said:

What do you know about integrals?

Not much. I know they can be used to find an area under a curve. I know a little about definite integrals. They show the area from one point to another. I know a little about Simpsons and trapezoid rules, but I don't know how to use them.

[studiot]

It would be helpful if you indicated what you mean by 'solve an integral'.

I suspect you are looking for what is known as 'the primitive' in what is known as the fundamental theorem of calculus.

 

[grayson]

4 minutes ago, studiot said:

It would be helpful if you indicated what you mean by 'solve an integral'.

I mean integrate something

edit: oops

Edited August 19, 2023 by grayson

[Genady]

2 minutes ago, grayson said:

Not much. I know they can be used to find an area under a curve. I know a little about definite integrals. They show the area from one point to another. I know a little about Simpsons and trapezoid rules, but I don't know how to use them.

I see. 

There are teachers here, but I am not. They might advise you where to start.

[studiot]

[math]\int_a^b {df=f(b) - f(a)} [/math]

 

Can you say what this means to you please ?

 

It is known as 'The Fundamental Theoem of Calculus'.

Edited August 19, 2023 by studiot

[grayson]

2 hours ago, studiot said:

∫badf=f(b)−f(a)

 

Can you say what this means to you please ?

 

It is known as 'The Fundamental Theoem of Calculus'.

Well, you are finding the definite integral from a to b of the derivative of f which equals f of b minus f of a

[studiot]

1 hour ago, grayson said:

Well, you are finding the definite integral from a to b of the derivative of f which equals f of b minus f of a

 

Exactly what I would expect your elementary Dummies book to say.

So can you explain it further, what do you think f is?

What do you think an integral actually is, when you say 'you are finding the (definite) integral....' ?

What are you expecting to find ?

 

These are not trick questions, they are meant to help you understand when I explain more fully.
This explanation will include the reason I am putting the word definite in brackets.

 

By the way the (definite) integral is not necessarily an area, though it can give you various area with suitable processing.

For example the (definite) integral of sin x from 0 to 2pi is zero so it will give you nothing at all.

 

 

 

 

 

[grayson]

2 minutes ago, studiot said:

 

Exactly what I would expect your elementary Dummies book to say.

So can you explain it further, what do you think f is?

What do you think an integral actually is, when you say 'you are finding the (definite) integral....' ?

What are you expecting to find ?

 

These are not trick questions, they are meant to help you understand when I explain more fully.
This explanation will include the reason I am putting the word definite in brackets.

 

By the way the (definite) integral is not necessarily an area, though it can give you various area with suitable processing.

For example the (definite) integral of sin x from 0 to 2pi is zero so it will give you nothing at all.

 

 

 

 

 

Well f is a function an integral usually finds the area under a curve but I remember more about indefinite than definite. Indefinite is jut reverse derivatives but I don't know how x^1/2(x-2) equals (x^3/2 - 2x^1/2) And when I say definite integral I mean antiderivative. I am expecting to learn calculus and Integrals. But the book stumped me at integrals.

[studiot]

23 minutes ago, grayson said:

Well f is a function an integral usually finds the area under a curve but I remember more about indefinite than definite. Indefinite is jut reverse derivatives but I don't know how x^1/2(x-2) equals (x^3/2 - 2x^1/2) And when I say definite integral I mean antiderivative. I am expecting to learn calculus and Integrals. But the book stumped me at integrals.

OK This is useful.

 

Yes f is a function. That is it is a whole set of values of y each one corresponding to a value of x.

Important features of functions are.

Each value of x must correspond to one and only one value of y

However many different values of x may correspond to the same value of y

For instance the constant function y = 3 makes y  to be 3 for each and every value of x.

The whole function is the whole set of pairs of values, one x and one y in each.  y = f(x) refers to all the pairs (x, y) and may represent a geometric curve.

If the function is not to broken up it may be 'differentiable'.

This means a second function called the derived function or just the derivative may be obtained from it by a process called differentiation.

Thus the derivative is a function, just like its progenitor.

 

The integral on the other hand is quite different. To integrate means to perform some sort of sum.
Indeed the symbol is a stylised Gothic S.

When you perform a sum you add things up and come up with a number.

A single number.

Luckily there are some formulae for performing this sum, but as Genady has already said, there is no general formule.

 

The Fundamental Theorem of Calculus also comes to our rescue when we want to find these sums ( hence the link to area, volume and many other things)

I called f'   the 'primitive' which gives rise to the derived function  - You can also use the term 'antiderivative' if you wish.

This is really wonderful news because it tells us that the number we want  -  the sum the integral represents  - is the difference between two numbers, the value of the primitive at each end on the region. I say region because the simple  formula integrates (sums) along a line but the region may be a line an area a volume or more.

A further comment looking ahead.

Finding the primitive allows us to 'solve' differential equations which are equations containing both algebraic functions and derivatives.
such equations arise quite naturally in just about every corner of science and engineering.

[grayson]

17 hours ago, studiot said:

OK This is useful.

 

Yes f is a function. That is it is a whole set of values of y each one corresponding to a value of x.

Important features of functions are.

Each value of x must correspond to one and only one value of y

However many different values of x may correspond to the same value of y

For instance the constant function y = 3 makes y  to be 3 for each and every value of x.

The whole function is the whole set of pairs of values, one x and one y in each.  y = f(x) refers to all the pairs (x, y) and may represent a geometric curve.

If the function is not to broken up it may be 'differentiable'.

This means a second function called the derived function or just the derivative may be obtained from it by a process called differentiation.

Thus the derivative is a function, just like its progenitor.

 

The integral on the other hand is quite different. To integrate means to perform some sort of sum.
Indeed the symbol is a stylised Gothic S.

When you perform a sum you add things up and come up with a number.

A single number.

Luckily there are some formulae for performing this sum, but as Genady has already said, there is no general formule.

 

The Fundamental Theorem of Calculus also comes to our rescue when we want to find these sums ( hence the link to area, volume and many other things)

I called f'   the 'primitive' which gives rise to the derived function  - You can also use the term 'antiderivative' if you wish.

This is really wonderful news because it tells us that the number we want  -  the sum the integral represents  - is the difference between two numbers, the value of the primitive at each end on the region. I say region because the simple  formula integrates (sums) along a line but the region may be a line an area a volume or more.

A further comment looking ahead.

Finding the primitive allows us to 'solve' differential equations which are equations containing both algebraic functions and derivatives.
such equations arise quite naturally in just about every corner of science and engineering.

I know what an antiderivative is. I am just having trouble solving it. Like for example, why does X^1/2(X-2)=(x^3/2-2x^1/2)?

[Genady]

Just now, grayson said:

Like for example, why does X^1/2(X-2)=(x^3/2-2x^1/2)?

X^1/2(X-2)=

X^1/2*X - X^1/2*2=

X^1/2+1 - 2X^1/2=

X^3/2 - 2X^1/2

It is algebra. Nothing to do with integrals.

[joigus]

5 minutes ago, grayson said:

I know what an antiderivative is. I am just having trouble solving it. Like for example, why does X^1/2(X-2)=(x^3/2-2x^1/2)?

Looks like you're having trouble with the properties of powers, not with antiderivatives.

Are you familiar with x^n+m=xⁿx^m?

x-posted with @Genady

[grayson]

Just now, joigus said:

Looks like you're having trouble with the properties of powers, not with antiderivatives.

Are you familiar with x^n+m=xⁿx^m?

x-posted with @Genady

Well in my calculus for dummies book, The power rule said that 5⁴ = 4*5³ I don't know how that has anything to do with the last equation I said. And yes I understand the concept of what you are saying

[joigus]

1 minute ago, grayson said:

Well in my calculus for dummies book, The power rule said that 5⁴ = 4*5³

Doubtful. That is plain wrong.

[grayson]

3 minutes ago, joigus said:

Doubtful. That is plain wrong.

When you say that is wrong are you saying in derivatives or antiderivatives? Because that was in the derivatives section

[Genady]

6 minutes ago, joigus said:

Doubtful. That is plain wrong.

It looks like misunderstanding of (xⁿ)' = n*x^n-1.

[joigus]

2 minutes ago, Genady said:

It looks like misunderstanding of (xⁿ)' = n*x^n-1.

That must be it.

[grayson]

9 minutes ago, joigus said:

That must be it.

So it has to be in parentheses first? which means it would be (5⁴) instead of 5⁴?

[joigus]

2 minutes ago, grayson said:

So it has to be in parentheses first? which means it would be (5⁴) instead of 5⁴?

No. It means you can't take x to be 5, or any other particular value. It must be a variable (varying, non-fixed) quantity. So the derivative of x⁵ is indeed 5x⁴, while 4*5⁴ 'is nothing of' 5⁵.

And the derivative of x⁵ at x=5 is indeed 4*5⁴

But that is not what you said...

It seems as if you're getting ahead of yourself. Maybe you need a good calculus book --like Spivak--, instead of calculus for dummies.