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Integration

Integration. The Explanation of integration techniques. There are lots of different types of integration but we are going to look at two integration techniques. Integration by parts. Integration by substitution. Integration By Parts.

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Integration

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  1. Integration The Explanation of integration techniques

  2. There are lots of different types of integration but we are going to look at two integration techniques. • Integration by parts. • Integration by substitution.

  3. Integration By Parts Let u and v be real-valued functions which are continuous on [a,b] and have continuous derivatives and . Then,

  4. Examples Using Integration By Parts Show, Which when put into the integration by parts formula will give

  5. We can simplify this by putting in our limits and using the identity To get,

  6. Hence using this proof we can solve

  7. Now using the following identities we can easily integrate this function.

  8. By adding the following equations, We obtain the following function, By substituting this in we get,

  9. Integration By Substitution This is the general equation for integration by substitution.

  10. There are three main steps to integration by substitution, Choose a suitable substitution. Integrate your chosen function and rearrange to get dx on its own. Change your limits.

  11. Examples Using Integration By Substitution. Example 1 Side workings By using the information gained in the side workings we obtain the following integral

  12. Example 2 Side workings By using the information gained in the side workings we obtain the following integral

  13. Example 3 Side workings Before we can use integration by substitution we need to complete the square. By using the information gained in the side workings we obtain the following integral For this example we need to use integration by substitution twice

  14. By using the information gained in the side workings we obtain the following integral Side workings To integrate this function we need to use the following identities. Which when added together with give us a function which can be easily integrated.

  15. When we put our limits in we get,

  16. Questions?

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