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:
In our case, the inner function:
and the outer function:
Plugging all these into our "integration by substitution" tool gives:
But this is slightly different to the original integral due to that minus sign. Of course this is trivial to deal with as:
Great! So now we have the structure we require to apply integration by substitution! We can substitute all of that with , and all the scary bits go away. So:
Then substituting (=) back in:
So looking at the first one from a few days ago (Oct 13th):
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,
But given the and the exponential, we'll also probably need to use integration by parts:
So, as above, letting:
Then plugging , , and into our lovely integration by parts tool above gives:
Which looks very familiar with what we've started with, except is now just ! If we can reduce by another power, we'll end up with just which will surely give us a much easier integral to solve.
So applying integration by parts again, but letting:
Putting these in a safe place for later. Found these to be a really good revision questions for integration.
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)