## Integrals with sec and tan when the power of tan is odd

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.…