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

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.…

Fundamental theorem of calculus example

We did an example today in class which I wanted to go through again here. The question was to calculate


\frac{d}{dx}\int_a^{x^4}\sec t dt


We spot the pattern immediately that it’s an FTC part 1 type question, but it’s not quite there yet. In the FTC part 1, the upper limit of the integral is just x, and not x^4. A question that we would be able to answer is:


\frac{d}{dx}\int_a^{x}\sec t dt


This would just be \sec x. Or, of course, we can show that in exactly the same way:


\frac{d}{du}\int_a^{u}\sec t dt=\sec u


That’s just changing the names of the variables, which is fine, right? But that’s not quite the question. So, how can we convert from x^4 to u? Well, how about a substitution? How about letting x^4=u and seeing what happens. This is actually just a chain rule. It’s like if I asked you to calculate:


\frac{d}{dx} g(x^4).


You would just say: Let x^4=u and then we have:


\frac{d}{dx} g(x^4)=\frac{du}{dx}\frac{d}{du}g(u)=4x^3 g'(u).…

Is MAM1000W Making You Anxious?

Hello, my name is Jeremy :-)

I am new to the MAM1000W team of tutors – if you want to read more about my background you can take a look at my bio in the MAM1000W document on Vula. In short, I returned to UCT last year to do my second undergraduate degree, a BSc in Applied Maths and Computer Science, at 25 years old, after not doing any maths for seven years. In the beginning, I found MAM1000W really hard; the pace of the content and the tutorials made me anxious and when test one came around I scored 50%. More anxiety. Luckily I have a great support system (inside and outside the Math department) and with some good advice and determination, I was able to figure out a new, way of studying and managing my time that worked better for me. When it came time for test 2, despite being super stressed out, I scored 81%.…

By | July 22nd, 2018|Uncategorized|1 Comment