Up Learn – A Level MATHs (Edexcel) – Calculus III

Integral of ln(x)

To find the integral of ln(x), use integration by parts. We came up with a game – the ‘one up, one down’ game – to show you exactly how to use integration by parts in all cases.

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
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
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

Last time, we saw that, when we integrate by parts, we’re really using…

When we integrate by parts, we’re really using the product rule for differentiation. 

Now, when we’re playing the one up, one down game, we normally differentiate the algebraic part.

Except, when… 

Except when the other part is logarithmic. [xln x ]

And that’s because, if we differentiate this part [x], we have to integrate this part [ln x ]

Which, actually, we haven’t seen how to do yet… 

But now, ironically, integration by parts allows us to find the integral of this function [ln x dx]

Now, looking at this expression, it doesn’t look like there are even two parts… 

But, actually, we could say that the two parts are… 

We could say that the two parts are ln x … and, a secret one! [reveal the secrets… 1⋅ln x ]

And so now, given these two parts, which should we differentiate, and which should we integrate? 

As usual, we should differentiate the log function, and integrate our secret 1 – as, whenever we have a log function, we always differentiate that! 

And so now, find the integral of ln x … 

First, we rewrite with integration by parts, like this [xln x -∫x⋅1xdx]

Then, this simplifies to just… 1…  [1dx]

Giving us this as our final integral [ln x dx≡xln x -x+c]

And so this time, integrate this [4ln x dx]

First, [click 1 time] we can rewrite this integral like this [4xln x -4x⋅1xdx]

Which simplifies to this…  [becomes 4 on right]

Finally, that gives us this integral [4ln x dx≡4xln x -4x+c]

So, to sum up, it’s possible to integrate natural log functions using integration by parts… 

For example, to integrate this [ln x ]… 

We differentiate the log function, and integrate… 

And integrate a secret one! 

Then, we just use integration by parts as normal!