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

Integration by Parts: Summary

Integration by parts is a trick for integrating many functions in the form f(x)g(x). It is a major part of integration at A Level.

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

We’ve now seen how to fully master the one up, one down game… 

For example, following the rules of the game, turn this into an integral that we can integrate [5xcos x dx]…

Differentiate this part, and integrate this one, to get this integral [5sin x dx]

And now that we’ve seen how to master that game, we’re on the verge of being able to integrate a whole bunch of more complex functions, like this one [example from above, 5xcos x dx]

Except, the only problem is, we definitely can’t just take an integral…and then differentiate one part and integrate the other to make it simpler… 

As that would turn it into a totally different integral!

But fortunately, there is [click 2 times – wait for the animations] something we can do … [Pause audio for 2 seconds]

So, to integrate this function [same example still]

First, [click 2 times – waiting for the animations to finish] play the game [Pause audio for 2-3 seconds] to get this simpler integral [5sin x dx, see below for position]

Then, take the function you differentiated…  [5x], and write it here… 

Next, look at the function you integrated… 

Take it’s integral, from here [sin x  from new integral, not including 5 as that came from 5x]… 

And write it here – so that it’s multiplied by this function… 

Finally, place a minus symbol here… 

And that’s it… this original integral, is identical to this…  [must be laid out as below]

5xcos x dx ≡ 5xsin x -5sin x dx

And even though we’ll still have to integrate this to fully find this integral [LHS]… 

This one’s much easier to integrate than this one!

And so next, playing the one up, one down game with this function [-9xsin x dx] gives us this integral [-9-cos x dx], which we can simplify [9cos x ]…but we’re not going to just yet… [revert to previous version]

So, if we want to rewrite this integral [first one, now lay screen out like below…] 

-9x sin x dx≡_________-(-9)(-cos x )dx

Then what do we need to write here?

Here, we need to put the part we differentiated… [click 1 time]  [-9x]

And then multiply it by the integral of the other part [Click 1 time] [-cos x  from integral]. [Pause audio for 2 seconds]

Giving us this [(-9x)(-cos x )], and then this [9xcos x ]

And now we can rewrite this integral too [click 1 time] [last one to get rid of negatives and brackets]

So, this time, playing the one up, one down game with this function gives us this integral…  

[6xsec2x dx  6tan x dx]

So, which of these is the correct way of rewriting this integral?

To rewrite this integral, we differentiate one part and integrate the other to get this integral… 

And then, we need to take the term we differentiated…

Multiply it by the integral of the term we integrated… 

And finally, subtract our new integral… 

Giving us this expression 

Now, rewriting integrals in this way is called integrating by parts… 

Because it involves manipulating these two parts… 

And so, we’ll practice integrating by parts in full….next…