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Integration - The Basics watch

1. Hi,

I understand how to get to the answer, that's fine. But I just don't understand the logic used. "Consider", what do they mean? Are we just "Letting" y=... or is y=sin(2x+3) the actual integral?

If so, if we differentiate this value, we should get back to cos(2x+3) but we clearly don't. So this value cannot be the integral? Why are we taking these steps?

Thanks!
Attached Images

2. (Original post by ps1265A)

Hi,

I understand how to get to the answer, that's fine. But I just don't understand the logic used. "Consider", what do they mean? Are we just "Letting" y=... or is y=sin(2x+3) the actual integral?

If so, if we differentiate this value, we should get back to cos(2x+3) but we clearly don't. So this value cannot be the integral? Why are we taking these steps?

Thanks!

If you differentiate that you get cos(2x+3)

The text is explaining why you you need the 1/2
3. (Original post by TenOfThem)

If you differentiate that you get cos(2x+3)

The text is explaining why you you need the 1/2
I want to know whether step 1 is just a "rough" integration

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4. (Original post by ps1265A)
I want to know whether step 1 is just a "rough" integration

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No

Step 1 is telling you the thinking process
5. (Original post by TenOfThem)
No

Step 1 is telling you the thinking process
And what is the thinking process? An estimation?

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6. (Original post by ps1265A)
And what is the thinking process? An estimation?

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It is looking at the integrated and saying to yourself ... Oh ... I must have differentiated a Sin function in order to get a Cos function

I am struggling to see what is confusing you
7. (Original post by ps1265A)
I want to know whether step 1 is just a "rough" integration

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Step 1 is an "intelligent " guess. You then check whether it is correct and adjust if necessary.
You should try to get to know certain standard patterns. For example, consider a function of x f(x). If the integral is F(x) then the integral of f(ax) will be F(ax)/a
Similarly the integral of f(ax+b) will be F(ax+b)/a.
IN reverse, if the derivative of f(x) is g(x) say then the derivative of f(ax) will be ag(ax) and of f(ax+b) will be ag(ax+b).
More generally the derivative of f(g(x)) will be g'(x)f'(g(x))
See page 38 in the attachment.
Attached Images
8. Notes for C1-C4.pdf (918.2 KB, 501 views)
9. (Original post by TenOfThem)
It is looking at the integrated and saying to yourself ... Oh ... I must have differentiated a Sin function in order to get a Cos function

I am struggling to see what is confusing you
Yes, that's exactly what I wanted to know and was thinking the same, just didn't know how to word it

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10. (Original post by brianeverit)
Step 1 is an "intelligent " guess. You then check whether it is correct and adjust if necessary.
You should try to get to know certain standard patterns. For example, consider a function of x f(x). If the integral is F(x) then the integral of f(ax) will be F(ax)/a
Similarly the integral of f(ax+b) will be F(ax+b)/a.
IN reverse, if the derivative of f(x) is g(x) say then the derivative of f(ax) will be ag(ax) and of f(ax+b) will be ag(ax+b).
More generally the derivative of f(g(x)) will be g'(x)f'(g(x))
See page 38 in the attachment.
Thank you soooo much, I understand it now

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11. (Original post by ps1265A)
Thank you soooo much, I understand it now

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You''re welcome
12. (Original post by brianeverit)
You''re welcome
Could you tell me what the best way to learn integration is? At the moment, I have 2 option: one is by simply looking at the formulas and just plugging in and one is by actually doing the steps as I did above. Or do I have to adopt a combination of both because all questions don't follow the above method?

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13. (Original post by TenOfThem)
It is looking at the integrated and saying to yourself ... Oh ... I must have differentiated a Sin function in order to get a Cos function

I am struggling to see what is confusing you
Could you tell me what the best way to learn integration is? At the moment, I have 2 option: one is by simply looking at the formulas and just plugging in and one is by actually doing the steps as I did above. Or do I have to adopt a combination of both because all questions don't follow the above method?

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14. (Original post by brianeverit)
You''re welcome
Because say if I have the Q was integrate 1/(2x+1)^2 and I followed the 3 step method, I would get:

But the actual answer is -1/2(2x+1)^2

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15. (Original post by ps1265A)
Because say if I have the Q was integrate 1/(2x+1)^2 and I followed the 3 step method, I would get:

But the actual answer is -1/2(2x+1)^2

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Both this question and the original question use the idea of "reverse chain rule"

You should know that differentiating the function of a function needs the chain rule

So if you are integrating the function of a function you will consider reversing the chain rule
16. (Original post by TenOfThem)
Both this question and the original question use the idea of "reverse chain rule"

You should know that differentiating the function of a function needs the chain rule

So if you are integrating the function of a function you will consider reversing the chain rule
I have used the reverse chain rules

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17. (Original post by ps1265A)
I have used the reverse chain rules

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Sorry did not look at your work properly

You need to increase the power

You have changed a power of -2 to a power of -3

That is decreasing
18. (Original post by TenOfThem)
Sorry did not look at your work properly

You need to increase the power

You have changed a power of -2 to a power of -3

That is decreasing
I'm sorry if the picture is unclear. I've changed my power from a 2 to a 3

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19. (Original post by TenOfThem)
Sorry did not look at your work properly

You need to increase the power

You have changed a power of -2 to a power of -3

That is decreasing
Or do I have to account for the reciprocal?

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20. (Original post by ps1265A)
Or do I have to account for the reciprocal?

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I can see that you have changed from a 2to a 3

As you have realised now ... You have changed -2 to -3 ... Decreasing rather than increasing ... Since the function is in the denominator
21. (Original post by TenOfThem)
I can see that you have changed from a 2to a 3

As you have realised now ... You have changed -2 to -3 ... Decreasing rather than increasing ... Since the function is in the denominator
So this is what I've got:

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