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

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

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. Next

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..... Next

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 ? Next

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: Next

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: Next

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 . Next

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 Next

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

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

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. Next

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: Next

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 . Next

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

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... Next

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 . Next

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. Next

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 Next

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: Next

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

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: Next

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: Next

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: Next

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. Next

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: Next

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 Next

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. Next