Up Learn – A Level MATHs (edexcel) – Calculus III
Identifying f'(x) Divided by f(x)
Before using the first reverse chain rule ‘trick’, you need to be able to identify functions in the form f'(x) divided by f(x).
More videos on Calculus III:
The Reverse Chain Rule: Summary
Identifying f'(x) Divided by f(x)
Integrating f'(x) Divided by f(x)
Integrating kf'(x) Divided by f(x): Part 1
Integrating Parametric Equations: Summary
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Calculus III
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
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
25. Finding Your Own Substitution
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
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
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?
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
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
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 that there are some functions that are possible to integrate … and some that are not possible to integrate
And that there is no chain rule, no product rule, and no quotient rule for integration
Meaning, to integrate more complex functions, we have to learn ways of using what we know about differentiation…backwards…
Now, the first method which allows us to do this [show the RCR box]…is actually more like a collection of closely related tricks…[show the three boxes below coming out of it, see below for guidance…]
And to start off, we’ll spend some time looking at the first two of these integration tricks.
However, before we can use the first of these tricks, we need to be able to recognise a special type of function…
Like this one, for example [12×2+54×3+5x]
Here, we can say that…
Here, we can say that this expression [top one]…
Is the derivative of this expression [bottom one]
Now, our first integration trick allows us to integrate any function like this…
One where the top of a fraction… is the derivative of the bottom…
So, in which of these is the top of the fraction… the derivative of the bottom?
For these fractions, the top of the fraction is the derivative of the bottom.
But for these, it is not.
And next, in which of these fractions is the numerator the derivative of the denominator?
For these fractions, differentiating the denominator, gives the numerator…
But for these, it does not.
And so finally, which of these fractions could we integrate using our first integration trick?
In each of these fractions, the numerator is the derivative of the denominator…
Meaning we could integrate each with our first trick for integration.
Now, we’ll see what this trick is… next…
But to sum up for now, our first integration trick allows us to integrate expressions of the form…
Our first integration trick allows us to integrate expressions of this form – where the top of a fraction is the derivative of the bottom. [click one time]