1. Integration by Substitution and by Parts
Department of Mathematics
University of Leicester

2. Integration by Substitution
Content
Introduction
Integration by Substitution
Integration by Parts
Another useful result

3. Integration by Substitution
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Introduction
Each function has its own integral (what it integrates to).
We differentiated complex functions using the Product, Substitution and Chain Rules.
With Integration, we will use ‘Integration by Parts’ and ‘Integration by Substitution’ to integrate more complex functions, using the simple functions as a starting point.
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4. Integration by Substitution: Introduction
Integration by Parts
Another useful result
Integration by Substitution: Introduction
Suppose we had to integrate something like:
We know how to integrate , so what we really want to do is turn into a new variable, .
Then we’ll have , which we know how to integrate.....
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5. Integration by Substitution: Introduction
Integration by Parts
Another useful result
Integration by Substitution: Introduction
But this isn’t the end of the story.
We’ve now got
This doesn’t make any sense because we’ve got and inside the integral. We need to change everything so we’ve only got .
How can we rewrite in terms of and ?
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6. Integration by Substitution: Introduction
Integration by Parts
Another useful result
Integration by Substitution: Introduction
Well, , so
Think of as a fraction, so we can rearrange to get
Put this back into the integral:
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7. Integration by Substitution: Introduction
Integration by Parts
Another useful result
Integration by Substitution: Introduction
Then all you have to do is write everything in terms of again:
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8. Integration by Substitution: General Method
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Substitution: General Method
Choose a part of the function to substitute as (choose a part that’s inside a function)
Find and rewrite in terms of
Rewrite everything inside the integral in terms of .
Do the integration.
Rewrite in terms of .
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9. Integration by Substitution: Example:
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Substitution: Example:
Integrate
let (because it’s inside the √-function.)
, so
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10. Integration by Substitution: Example
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Substitution: Example
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11. Choose the appropriate substitutions:
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Choose the appropriate substitutions:
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12. Integration by Substitution: Working with Limits
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Substitution: Working with Limits
If there are limits on the integral, then you have 2 options:
Ignore the limits, work out the integral, then put them back in at the very end.
Rewrite the limits in terms of u, then just work with u and don’t change back to x at the end.
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13. Integration by Substitution: Working with Limits
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Substitution: Working with Limits
Consider the previous example:
Ignore the limits: We get as before.
Then put the limits in to get:
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14. Integration by Substitution: Working with Limits
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Substitution: Working with Limits
Change the limits (using )
upper limit:
lower limit:
Then just forget and integrate for .
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15. Integration by Substitution: Question
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Substitution: Question
Calculate
_____
105
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16. Integration by Parts: Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Parts: Introduction
Suppose we had to integrate something like:
We know how to integrate and , but not when they are multiplied together.
We have to use the following rule to integrate a product...
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17. Integration by Parts: The formula
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Parts: The formula
are the two functions that form the product.
is what you get if you integrate
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18. Integration by Parts: The method
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Parts: The method
Decide which function you want to differentiate (this will be ) and which you want to integrate (this will be )
Find and from
Put these into the formula.
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19. Integration by Substitution
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Parts: Choose the best way to split these integrals into parts...
pi ln 5 (1st one, forgot to square f(x), 2nd one, did it round y-axis).
6 pi (2nd one, did from 0 to 4, 3rd one, forgot to square g(y))
3pi (2nd one, did y^2 – 1 not (y-1)^2, 3rd one, did it round x-axis
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20. Integration by Parts: Example
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Parts: Example
Integrate
Differentiate because this will give a nice simple constant.
So the integral is:
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21. Integration by Substitution
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Parts
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22. Integration by Parts: Proof of formula
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Parts: Proof of formula
To prove the formula, we start with the product rule for differentiation:
Integrate both sides:
Rearrange:
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23. Integration by Parts: Another example
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Parts: Another example
Integrate
It doesn’t seem to matter which function we differentiate and which we integrate.
Let , so
Then the integral is:
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24. Integration by Parts: Another example
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Parts: Another example So
Choose the same way round as last time (otherwise we’ll reverse what we’ve just done).
Let , so
Then we get
We perform Integration by Parts again on this term:
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25. Integration by Parts: Another example
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Integration by Parts: Another example
So:
ie.
If you can’t see what to do straight away, you should just play around with an integral until you spot something that works.
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26. Integration by Substitution
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Another useful result
We prove it using integration by substitution:
Let We get , so
So the integral becomes:
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27. Decide which statements are true:
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Decide which statements are true:
(red is false, green is true)
T,T,F,F,F,,T,F,T
shapes 1 to 6 are the red text boxes
shapes 7 to 12 are the green text boxes
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28. Integration by Substitution
Introduction
Integration by Substitution
Integration by Parts
Another useful result
Conclusion
We can integrate simple functions using the rules for those functions.
We can integrate more complex functions by substituting for part of the function, or by integrating a product.
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