The Reverse Chain Rule: Summary

The term ‘reverse chain rule’ refers to a set of tricks for integrating more complex functions. At A Level, there are a few tricks to know.

Up Learn – A Level Maths (edexcel)

Calculus III

1. Introduction to Integrating More Functions
2. The One Algebraic Function that Got Away
3. The Purpose of the Modulus
4. What Went Wrong?
5. What We Know So Far
6. Integrating Exponential Functions
7. Integrating Trig Functions
8. Integrating sin(x)
9. Integrating cos(x)
10. Integrating tan(x)
11. Integrating cot(x)
12. Integrating cosec(x)
13. Integrating sec(x)
14. Integrating Four More Trig Functions
15. Integrating sec^2(x)
16. Integrating cosec^2(x)
17. Integrating cosec(x)cot(x)
18. Integrating sec(x)tan(x)
19. Revisiting Definite Integrals
20. The Area Under a Sine Curve
1. Introduction to the Reverse Chain Rule
2. What We Can Differentiate
3. Liouville’s Vision
4. The Fate of Integration
5. Identifying f'(x) Divided by f(x)

6. Integrating f'(x) Divided by f(x)
7. Integrating kf'(x) Divided by f(x) Part 1
8. Identifying kf’x Divided by f(x)
9. Integrating kf'(x) Divided by f(x) Part 2
10. Integration and Partial Fractions
11. Identifying f'(x) Multiplied by (f(x))^n
12. Identifying kf'(x) multiplied by (f(x))^n
13. Integrating kf'(x) Multiplied by (f(x))^n
14. The Reverse Chain Rule Part 1
15. The Reverse Chain Rule Part 2
16. Difficulty with the Reverse Chain Rule
17. Making a Substitution
18. Converting the Infinitesimal
19. Integrating with Respect to u
20. Integration by Substitution
21. Non-Linear Substitutions
22. Speeding Up the Process
23. Substitutions Where x is the Subject
24. Implicitly Defined Substitutions
26. Pick the Expression That’s Been Raised to a Power
27. Why Did We Learn the First Two Tricks?
28. Converting the Boundaries of a Definite Integral
1. Introduction to Integration by Parts
2. The One Up, One Down Game
3. Mastering the One Up, One Down Game
4. Changing the Rules of the Game
5. When Both Expressions are Algebraic
6. When One Expression is Logarithmic
7. Integration by Parts 1
8. Integration by Parts 2
9. Completing Integration by Parts
10. Integrating by Parts Multiple Times
11. Integration by Parts and the Product Rule
12. Integrating ln(x)
13. Integration by Parts and Definite Integrals
1. Introduction to Integrating More Trigonometric Functions
2. The Return of the Identities
3. A Reciprocal Pythagorean Identity
4. Another Reciprocal Pythagorean Identity
5. The Original Pythagorean Identity
6. The Double Angle Identity for Sine
7. The Double Angle Identity for Cosine Part 1
8. The Double Angle Identity for Cosine Part 2
9. The Double Angle Identity for Cosine Part 3
10. The Double Angle Identity for Tangent
11. What Trig Functions Can We Now Integrate?
1. Introduction to Finding More Areas with Integration
2. Finding the Area Between Two Curves
3. Finding the Area Between Two Points of Intersection
4. The Areas We Can’t Find Yet
5. What is a Trapezium?
6. Finding the Area of a Trapezium
7. Splitting an Area into Right Trapeziums
8. Finding the Width of the Strips
9. Finding the Boundary Points
10. Finding the Value of y at Each Boundary Point
11. Finding the Area Under the Curve
12. The Trapezium Rule Part I
13. The Trapezium Rule Part II
14. The Trapezium Rule Part III
15. The Trapezium Rule Part IV
16. Overestimating and Underestimating the Area
1. Integration and Parametric Equations
2. Convert to Cartesian Form then Integrate
3. Integrate Without Converting to Cartesian Form
4. Rewriting the Integral in Terms of x
5. Rewriting the Boundaries in Terms of t
6. Recapping the Strategies
1. Introduction to Differential Equations
2. Differential Equations in the Real World
3. Difficulties with Differential Equations
4. The Constant of Integration and Families of Curves
5. Methods for Differential Equations
6. Integrating as Normal
7. Solving Differential Equations
8. Why do we Use the Term ‘Solution’
9. The Differential Equations We Can Solve So Far
10. Recognising a Special Type of Differential Equation
11. Separation of Variables Part 1
12. Separation of Variables Part 2
13. Finding Particular Solutions to Differential Equations
14. Modelling with Differential Equations I
15. Modelling with Differential Equations II
16. Modelling with Differential Equations III

We’ve now seen two special integration tricks.

The first allows us to integrate an expression like this, to find…  [3cos x sin x +1dx]

To find that this is the integral.

The second allows us to integrate an expression like this, to find…  [sec2xtan6x dx]

To find that this is the integral.

Now, we can show that each of these results is correct…

And more broadly, that our two tricks definitely work…

By taking these expressions [integrals], and differentiating them…

So, starting with this one [3ln sin x +1 +c]…

And…ignoring the mod bars [fade them] and the + c [fade them]…

To differentiate this expression [3ln⁡(sin(x)+1)], we’ll have to use…

We’ll have to use the chain rule, making our inner function, u, equal to…

Making our inner function, u, equal to sin x +1, and then differentiating, to get…

To get this derivative

– the expression we started with!

That’s why the first integration trick always works!

We know that, if we ever see an expression like this [bottom], its integral must look like this [original function we differentiated]

Last time, we took two integrals [from start of last script]

And used the first to show why the first integration trick always works…

And so next, we can do the same for the second integration trick, using this result [17tan7x +c]…

First, we can ignore the plus c like before…

And then, to differentiate, we need to use…

Next, to differentiate, we need to use the chain rule again, setting u equal to…

Setting u equal to tan x .

So, with u equal to tan x , we can differentiate to find that…

And actually, whenever we have a function raised to a power… [y=17u7]

Then, regardless of what the function is [tan x ]…

This derivative we get will always be of this form… [maybe show dydx=u6dudx somehow, using working above]

With the function, raised to a lower power, multiplied by its derivative.

That’s why the second trick works!

So, both tricks take a special case of the chain rule…

For which we know what the derivative will look like…

And say, ‘okay, if you see an expression like this [highlight bottom], which we could have got after differentiating an expression like this [top two] with the chain rule…

‘Then we know what the integrals of functions like these must look like’ [maybe swap order of rows below as well, although make it clear that you’re doing that, if you do]

And so, both tricks are allowing us to use the chain rule…in reverse…

Meaning, whenever we use either of these two tricks…

We say we’re using the reverse chain rule.

That’s the general method we’re looking at here [emphasise the RCR box], a collection of ways of using the chain rule…backwards…

… although, arguably, the reverse chain rule isn’t really a rule … it just says that it’s sometimes possible to use the chain rule backwards!

So, to sum up, our first two integration tricks work because of the chain rule.

And that’s because, whenever we differentiate expressions of these forms…  [ln |fx|  and fxn]

We get expressions in these forms [f’xfx and f’xfxn-1]

So, whenever we use either of these first two integration tricks, we say that we’re using the…

We say that we’re using the reverse chain rule.