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# Trig integration watch

1. What method should I use to integrate sinxcos^2x
It seems like it's going to get very complicated with any method
2. by substitution, can you find an appropriate value for u?
3. (Original post by G.Y)
What method should I use to integrate sinxcos^2x
It seems like it's going to get very complicated with any method
What does differentiate to?
4. (Original post by Bobjim12)
by substitution, can you find an appropriate value for u?
Took me a very long time to find the right one. Usually with questions like this first thing to do is use a trig identity to change the cos^2x into cos2x. With this one cos^2x was the substitution. Very unexpected. Any tips on how to spot this quicker?
5. (Original post by G.Y)
Took me a very long time to find the right one. Usually with questions like this first thing to do is use a trig identity to change the cos^2x into cos2x. With this one cos^2x was the substitution. Very unexpected. Any tips on how to spot this quicker?
You didn't need to do any substitution. When I did A Level C4, the method of Inverse Chain Rule was in the textbooks that you can simply recognise that the integral of f'(x)f^(n)(x) dx=f^(n+1)(x)/(n+1)+(C) for n not -1 and if n=-1 then you get integral f'(x)/f(x) dx =ln(f(x))+(C) and you can easily prove those result by differentiating.

Here you had the integral of sin(x)cos^2(x) dx = - integral -sin(x)cos^2(x) dx= - integral f'(x)f^2(x) dx where f(x)=cos(x) and thus the integral is just -cos^3(x)/3+(C).

However, it interests me that the simple method of Inverse Chain Rule is so underused. At University, they'll always do the substitution wasting their time doing what should be trivial integrals based on the results even in a C4 textbook.

To add to this, it is so much easier to do many integrations if you use the Inverse Chain Rule as it informs you of what integrals can easily be done like when calculating more complicated integrals you would need to separate out those integrals where possible and also when deriving many reduction formulas.
6. (Original post by G.Y)
Took me a very long time to find the right one. Usually with questions like this first thing to do is use a trig identity to change the cos^2x into cos2x. With this one cos^2x was the substitution. Very unexpected. Any tips on how to spot this quicker?
I'm not sure how you went about with that substitution but u = cosx would be much easier.

And to everyone else, for me C4 didn't teach inverse chain rule, although my fp1 teacher decided to teach me it, and then it came very helpful in fp2.
7. (Original post by Dalek1099)
You didn't need to do any substitution. When I did A Level C4, the method of Inverse Chain Rule was in the textbooks that you can simply recognise that the integral of f'(x)f^(n)(x) dx=f^(n+1)(x)/(n+1)+(C) for n not -1 and if n=-1 then you get integral f'(x)/f(x) dx =ln(f(x))+(C) and you can easily prove those result by differentiating.

Here you had the integral of sin(x)cos^2(x) dx = - integral -sin(x)cos^2(x) dx= - integral f'(x)f^2(x) dx where f(x)=cos(x) and thus the integral is just -cos^3(x)/3+(C).

However, it interests me that the simple method of Inverse Chain Rule is so underused. At University, they'll always do the substitution wasting their time doing what should be trivial integrals based on the results even in a C4 textbook.

To add to this, it is so much easier to do many integrations if you use the Inverse Chain Rule as it informs you of what integrals can easily be done like when calculating more complicated integrals you would need to separate out those integrals where possible and also when deriving many reduction formulas.
I'm familiar with the inverse chain rule but never used it with trig integrals as I didn't think it was possible/allowed. And I have no idea what that maths you've typed means^^
8. (Original post by Dalek1099)
You didn't need to do any substitution. When I did A Level C4, the method of Inverse Chain Rule was in the textbooks that you can simply recognise that the integral of f'(x)f^(n)(x) dx=f^(n+1)(x)/(n+1)+(C) for n not -1 and if n=-1 then you get integral f'(x)/f(x) dx =ln(f(x))+(C) and you can easily prove those result by differentiating.

Here you had the integral of sin(x)cos^2(x) dx = - integral -sin(x)cos^2(x) dx= - integral f'(x)f^2(x) dx where f(x)=cos(x) and thus the integral is just -cos^3(x)/3+(C).

However, it interests me that the simple method of Inverse Chain Rule is so underused. At University, they'll always do the substitution wasting their time doing what should be trivial integrals based on the results even in a C4 textbook.

To add to this, it is so much easier to do many integrations if you use the Inverse Chain Rule as it informs you of what integrals can easily be done like when calculating more complicated integrals you would need to separate out those integrals where possible and also when deriving many reduction formulas.
There is no formal technique called 'The Reverse Chain Rule', as far as I am aware. In reality, it is simply becoming adept enough with the method of substitution that one can evaluate the integral without needing to use substitution.

We cannot integrate functions except for a select few, which can be integrated using the Fundamental Theorem of Calculus. This is why the method of substitution exists, so we can transform any function into a form which can be integrated. And as above, we get the Reverse Chain Rule by just recognising how the substitution would work without having to do it.
9. (Original post by Dalek1099)
Here you had the integral of sin(x)cos^2(x) dx = - integral -sin(x)cos^2(x) dx= - integral f'(x)f^2(x) dx where f(x)=cos(x) and thus the integral is just -cos^3(x)/3+(C).
Could you explain this a bit more please
10. (Original post by MR1999)
There is no formal technique called 'The Reverse Chain Rule', as far as I am aware. In reality, it is simply becoming adept enough with the method of substitution that one can evaluate the integral without needing to use substitution.

We cannot integrate functions except for a select few, which can be integrated using the Fundamental Theorem of Calculus. This is why the method of substitution exists, so we can transform any function into a form which can be integrated. And as above, we get the Reverse Chain Rule by just recognising how the substitution would work without having to do it.

I've attached the C4 image of the Reverse Chain Rule students are expected to know. Bizarrely, it doesn't actually list the patterns that you are expected to recognise as Reverse Chain Rule when they quite clearly are. I am not certain whether that content is still in the C4 syllabus now though but I'd imagine it would be or at least is very useful to know.

Using Reverse Chain Rule doesn't require any knowledge of substitution,although it does kind of give justification for substitution, it uses the rules of differentiation and recognises that integration is the reverse of differentiation rather than getting students to simply use integration rules without any thought as to why what they are doing makes sense.

That's why I think that Reverse Chain Rule is so important to teach to students as it is teaching them to understand integration rather than rote learn techniques.
11. (Original post by G.Y)
Could you explain this a bit more please
Essentially, you need to recognise the rules given under section 5 there. If you don't understand why the formula differentiate the formula instead remembering the Chain Rule. Here we have f(x)=cos(x) and n=2.You do also need to take a minus outside of the integral to get it into that form as d/dx cos(x)=-sin(x) rather than sin(x).

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