We went through an example in class today which was

\[\int tan^6\theta \sec^4\theta d\theta\]

In this case we took out two powers of sec and then converted all the other \(\sec\) into $latex\ tan$, which left a function of tan times \(sec^2\theta d\theta\). We wanted to do this because the derivative of \(\tan\) is \(\sec^2\) and so we can do a simple substitution. If we have an odd power of \(\tan\), we can employ a different trick. Let’s look at:

\(I=\int \tan^5\theta\sec^7\theta d\theta\).

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 \(\sec\) and have only a factor which is the derivative of \(\sec\) left over. The derivative of \(\sec\) is \(\sec\tan\), so let’s try and take this out:

\(I=\int \tan^5\theta\sec^7\theta d\theta=\int \tan^4\theta\sec^6\theta (\sec\theta\tan\theta)d\theta\).

Now convert the \(\tan\) into \(\sec\) by \(\tan^2\theta=\sec^2\theta-1\):

\[I=\int (\sec^2\theta-1)^2\sec^6\theta (\sec\theta\tan\theta)d\theta=\int (\sec^{10}\theta-2\sec^8\theta+\sec^6\theta) (\sec\theta\tan\theta)d\theta\]

where here we have just expanded out the bracket and multiplied everything out.…