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,
Integration by parts is technique which can be used to integrate the product of two simpler functions. It is useful in many cases where other techniques will not work, although it cannot be used for all functions.
With this technique you have to split the function into two parts, integrate one part and differentiate the other. The choice of how to divide up the integral between u and dv is a matter of simply looking at the function, Usually, u is a simple function, such as a linear function of x, which becomes even simpler when differentiated.
However, when the integral involves a logarithm, this has to be the ‘u’ as ln x can’t be integrated easily.
If we allow u and v to be real values functions, which are continuous on the limits of a and b, and have continuous derivatives, we obtain the general formula for integration by parts.

4. Examples Using Integration By Parts
Show,
Which can then be used to prove the following question.
If we split the function into u and dv, integrating the dv function and differentiating the u function and the putting our new functions into the general formula on the pervious slide we get the following function.
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,
Which can be simplified by putting in the limits and using the identity sin^2x = 1 – cos^2x.
Which once rearranged will give us the what we were trying to prove.

6. Hence using this proof we can solve
Using what we have just proved we can solve the integral by subbing in the number 10 as our n value.

7. Which we can keep doing until we come to cos^2x, as this is a function we are able to integrate using the identities.
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,
By adding them together they give us a substitute for cos^2x , which we can then simply replace cos^2x with and ingrate the new function though normal method of integration to obtain an answer.

9. Integration By Substitution
This is the general equation for integration by substitution.
The chain rule allows you to differentiate a function of x by making a substitution of another variable t, integration by substitution is the corresponding integration method.
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.
There are three main steps to integration by substitution.
First you have to choose a suitable function for t to substitute
Then you have to integrate that function in order to obtain dt and substitute this in for dx, as you no longer want to integrate with respect to x but you now what to integrate with respect to t.
Finally you have to change your limits by placing your x values in the function you choose to equal t, as these to are in the form of x and need to been changed to t.
This is shown in the following examples.

11. Examples Using Integration By Substitution.
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
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,