Integration Revision - Integration By Substitution

Now, looking at the other integral from Oct 13th...


In my opinion, this was scarier than the first, but looking at it again, what makes it scary is the various powers. e to the power of x to the power of 2 and such.

But why should this be scary? There's practically only one tool that one could use to solve this: the only tool that includes functions of functions, integration by substitution:

\int f(g(x))\:g^{\prime}(x)\:dx=\int f(u)\: du

In our case, the inner function:

and the outer function:

Plugging all these into our "integration by substitution" tool gives:

\int -4xe^{-2x^{2}}\:dx

But this is slightly different to the original integral due to that minus sign. Of course this is trivial to deal with as:

\int^{x}_{0}4xe^{-2x^{2}}\:dx=-\int^{x}_{0} -4xe^{-2x^{2}}\:dx

Great! So now we have the structure we require to apply integration by substitution! We can substitute all of that with f(u), and all the scary bits go away. So:


=-\int^{x}_{0} -4xe^{-2x^{2}}\:dx

=-\int^{u}_{0} e^{u}\:du

=-[ e^{u}]^{u}_{0}



Then substituting u (=g(x)) back in:


Integration Revision - Integration By Parts, Twice

So looking at the first one from a few days ago (Oct 13th):

\int^{\infty}_{0}x^{2}\:\lambda\: e^{-\lambda x}\:dx=\frac{2}{\lambda^{2}}

I was totally lost with this. But running through some old tricks made this seem a lot more approachable. First off, moving the constant out brings a bit more clarity to the integrand,

\int^{\infty}_{0}x^{2}\lambda e^{-\lambda x}\:dx=\lambda\int^{\infty}_{0}x^{2} e^{-\lambda x}\:dx

But given the x^{2} and the exponential, we'll also probably need to use integration by parts:

\int^{b}_{a} f(x)g^{\prime}(x)dx=[f(x)g(x)]^{b}_{a}-\int^{b}_{a} f^{\prime}(x)g(x)dx

So, as above, letting:
g^{\prime}(x)=e^{-\lambda x},

we have:
g(x)=-\frac{1}{\lambda}e^{-\lambda x}

Then plugging f, f^{\prime}, g and g^{\prime} into our lovely integration by parts tool above gives:

\lambda\int^{\infty}_{0}x^{2}\: e^{-\lambda x}\:dx

=\lambda\left(\left[-x^{2}\: \frac{1}{\lambda}\:e^{-\lambda x}\right]^{\infty}_{0}-\int^{\infty}_{0}-2x\:\frac{1}{\lambda}\: e^{-\lambda x}\:dx\right)

=\lambda\left(0-0+2\frac{1}{\lambda}\int^{\infty}_{0}x\: e^{-\lambda x}\:dx\right)

=2\int^{\infty}_{0}x\: e^{-\lambda x}\:dx

Which looks very familiar with what we've started with, except x^{2} is now just x! If we can reduce x by another power, we'll end up with just 1 which will surely give us a much easier integral to solve.

So applying integration by parts again, but letting:
g^{\prime}(x)=e^{-\lambda x},

we have:
g(x)=-\frac{1}{\lambda}e^{-\lambda x}


2\int^{\infty}_{0}x\: e^{-\lambda x}\:dx

=2\left(\left[-x\: \frac{1}{\lambda}\:e^{-\lambda x}\right]^{\infty}_{0}-\int^{\infty}_{0}-\frac{1}{\lambda}\: e^{-\lambda x}\:dx\right)

=2\left(0-0+\frac{1}{\lambda}\int^{\infty}_{0} e^{-\lambda x}\:dx\right)

=\frac{2}{\lambda}\int^{\infty}_{0} e^{-\lambda x}\:dx

=\frac{2}{\lambda}\left[-\frac{1}{\lambda}\:e^{-\lambda x}\right]^{\infty}_{0}




End of Book 1 - Review of Understanding

A quick look at the mind map I produced for the whole of Book 1, very much a foundation for the rest of the materials on the module.

The complexity has increased necessarily, but I'd like to think the clutter has been reduced to a minimum. I was rearranging nodes as I was adding them in the hope of reducing clutter. You'll notice there are two white floating boxes not connected to anything externally. This was just to avoid clutter.

Some things to note about it that help me refer back to it:

It's split roughly into thirds, vertically. Each third is more or less a sub topic. More like themes, perhaps. Far left is foundational principles. Middle relates to basics of distributions. Far right is concepts surrounding the C.D.F. and P.D.F. (the definitions of which are in the white box in the middle of the far-right section).

Colouring helps a lot when needing to refer back to it, I found. Axioms are pink, properties are green, and definitions are blue. I found that in referring back to it, I needed some kind of differentiation between the definition of a certain distribution, and a normal definition. So all the yellow nodes you see are definitions of distributions.

In summary, to assist my understanding, I now have a graph that leads me from the most basics concept (like what an "event" is), to the definition of the standard normal distribution. I'll be referring back to this as I go!

Integration Revision

Putting these in a safe place for later. Found these to be a really good revision questions for integration.

\int^{\infty}_{0}x^{2}\lambda e^{-\lambda x}\:dx=\frac{2}{\lambda^{2}}



For context, the first one formed part of a question that required me to find the variance of a random variable, and the second one involved having to find the c.d.f. from a p.d.f.

(for my own reference, this was Book 1, Activity 5.2, p. 54 and Activity 6.1, p.62)

Current Application of Mind Maps

I've finally completed the second section of the first book, and it's served as a good revision/introduction session. I'm glad I went to the trouble of creating a mind map for the main concepts that were covered. Here's how my mind map grew as I added to it:



I've used this new mind map to refer back to several times during the exercises I've been set, and even the first assignment question. Admittedly it takes longer to get through all the learning material, as you have to take time out to contribute to the mind map as you go. Though I would say the time spent is worth it.

I also felt that this last image was the maximum size a mind map could take without becoming too cluttered. For the next section of the book, I'll be starting a brand new mind map. I'll try to keep going with this method for as long as I can, but it seems it's very useful.