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www.le.ac.uk Integration by Substitution and by Parts Department of Mathematics University of Leicester

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Content Integration by PartsIntegration by SubstitutionAnother useful resultIntroduction

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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. Next Integration by Parts Integration by Substitution Another useful result Introduction

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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 well have, which we know how to integrate..... Next Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Substitution: Introduction But this isnt the end of the story. Weve now got This doesnt make any sense because weve got and inside the integral. We need to change everything so weve only got. How can we rewrite in terms of and ? Next Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Substitution: Introduction Well,, so Think of as a fraction, so we can rearrange to get. Put this back into the integral: Next Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Substitution: Introduction Then all you have to do is write everything in terms of again: Next Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Substitution: General Method Next 1.Choose a part of the function to substitute as. (choose a part thats inside a function) 2.Find and rewrite in terms of. 3.Rewrite everything inside the integral in terms of. 4.Do the integration. 5.Rewrite in terms of. Integration by Parts Integration by Substitution Another useful result Introduction

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1.let (because its inside the -function. ) 2., so 3. Next Integration by Substitution: Example: Integrate Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Substitution: Example 4. 5. Next Integration by Parts Integration by Substitution Another useful result Introduction

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Choose the appropriate substitutions: Next Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Substitution: Working with Limits If there are limits on the integral, then you have 2 options: 1.Ignore the limits, work out the integral, then put them back in at the very end. 2.Rewrite the limits in terms of u, then just work with u and dont change back to x at the end. Next Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Substitution: Working with Limits Consider the previous example: 1.Ignore the limits: We get as before. Then put the limits in to get: Next Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Substitution: Working with Limits 2.Change the limits (using ) upper limit: lower limit: Then just forget and integrate for. Next Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Substitution: Question Next Calculate _____ 105 Integration by Parts Integration by Substitution Another useful result Introduction

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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... Next Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Parts: The formula are the two functions that form the product. is what you get if you integrate. Next Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Parts: The method 1.Decide which function you want to differentiate (this will be ) and which you want to integrate (this will be ) 2.Find and from. 3.Put these into the formula. Next Integration by Parts Integration by Substitution Another useful result Introduction

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Next Integration by Parts: Choose the best way to split these integrals into parts... Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Parts: Example 1.Differentiate because this will give a nice simple constant. 2. 3. So the integral is: Next Integrate Integration by Parts Integration by Substitution Another useful result Introduction

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Next Integration by Parts Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Parts: Proof of formula To prove the formula, we start with the product rule for differentiation: Integrate both sides: Rearrange: Next Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Parts: Another example 1.It doesnt seem to matter which function we differentiate and which we integrate. 2.Let, so 3.Then the integral is: Next Integrate Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Parts: Another example So 1.Choose the same way round as last time (otherwise well reverse what weve just done). 2.Let, so 3.Then we get Next We perform Integration by Parts again on this term: Integration by Parts Integration by Substitution Another useful result Introduction

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Integration by Parts: Another example So: ie. If you cant see what to do straight away, you should just play around with an integral until you spot something that works. Next Integration by Parts Integration by Substitution Another useful result Introduction

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Another useful result We prove it using integration by substitution: Let. We get, so So the integral becomes: Next Integration by Parts Integration by Substitution Another useful result Introduction

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Decide which statements are true: Next Integration by Parts Integration by Substitution Another useful result Introduction (red is false, green is true)

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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. Next Integration by Parts Integration by Substitution Another useful result Introduction

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