_{Up Learn – A Level MATHs (Edexcel) – Calculus III}

_{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.**

### 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

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!