Edexcel A2 C4 Mathematics June 2016 - Official Thread

Riemann sums actually don't converge if the function is unbounded on the interval. In order to define the integral, you have to split the interval up and take limits at every asymptote, like here http://www.sosmath.com/calculus/improper/intro/intro.html
I was thinking a little about this yesterday, the reason the Riemann sum fails seems to be because the Riemann sum allows for overestimation of the area, or more exactly, no matter how small the partition, we can choose a value of the function in the unbounded section as large as we want, making the area as large as we want. But not all this area lies under the curve.

Just a suggestion, I personally don't have an issue with it but you might want to move this conversation. Since this is the C4 thread, some people might not realise that these are questions that are well beyond the level of c4 at times and cause them to worry that they aren't able to answer them. Just something to consider, nice questions though I enjoyed them.

All questions here (if not most of them) are accessible with C4 knowledge, by no means are they easy or questions that you'll find in an Edexcel exam, but its good for people to try them. Though I definitely understand where you're coming from

Quick Q

In which situations is it sensible to take out numbers out of integrals, like 1/ln2 in this example?

If you've got any integral of the form $\displaystyle \int \lambda f(x) \, \mathrm{d}x$ then that is equivalent to:



for all integrable $f$ and all $\lambda \in \mathbb{R}$.

Some examples for clarification:

$\displaystyle \int 2(x+1) \, \mathrm{d}x = 2\int (x+1) \, \mathrm{d}x$

$\displaystyle \int 2x + 1 \, \mathrm{d}x = 2\int x + \frac{1}{2} \, \mathrm{d}x \neq 2 \int x+1 \, \mathrm{d}x$

PRSOM
(All that latex I can see when I quote)

Thanks, that makes a lot more sense now. I'll definitely try to practice more integration with that in mind now!

It looks like English to me.

No problem; it's certainly very useful for simplifying things. It's also quite nice (not sure if this is C4 or not, apologies if it isn't), let's say I want to integrate this:



I know that it looks very similar to a $f'(x)e^{f(x)}$ form which integrates to $e^{f(x)}$ (you know why, right?) but it's not quite. Instead we have $\frac{\mathrm{d}}{\mathrm{d}x} (6x^2) = 12x$, so what I really want is a $12xe^{6x^2}$, but that's not what I have. Instead of crying over that and being sad, let's pop in the $12$ there because we're badass (but also fix our damage, because we're nice like that!), what I mean is:



If you want to have a go, have a stab at $\int \frac{1}{1 + 5x} \, \mathrm{d}x$. We know that it's almost of the form $\frac{f'(x)}{f(x)}$ which integrates nicely to $\ln f(x)$, but it's not quite in that form, what do you need to do to get it in that form?

Edit to add: sorry if you don't care about any of this and I'm just waffling on like an idiot.

Thanks so much! My teacher actually just went over this in our last maths lesson and I kind of get it
I'll try that question some time later, need to prep for a bio test

*

(1/5)ln(1+5x)
I do it quicker with logs for some reason

2ln(x^2 + 3x)

Awesome, good stuff. I'll leave you alone now.

Haha, thanks

Rather do this than look blankly at my biology textbook

Female menstrual cycle...riveting

Does anyone have any advice or tips that could help?

Thanks guys

We'll make a mathematician out of you yet.

Sounds fascinating, I wouldn't want to know so much about my own body.

You have menstrual cycles?

I thought we were in sync.

We totally should be

I thought we were in sync.