We went through an example in class today which was
In this case we took out two powers of sec and then converted all the other into $latex\ tan$, which left a function of tan times . We wanted to do this because the derivative of is and so we can do a simple substitution. If we have an odd power of , we can employ a different trick. Let’s look at:
Here, sec is an odd power and so we can’t employ the same trick as before. Now we want to convert everything to a function of and have only a factor which is the derivative of left over. The derivative of is , so let’s try and take this out:
Now convert the into by :
where here we have just expanded out the bracket and multiplied everything out.
Now let’s use the substitution . Plugging this in, we get:
And in general this will always be the pattern we use when come across an integral like this with an odd power of . Look at the following example and see if you can get this: