College techniques for 'A'-Level/high school math - Part 1

While I was in high school, I observed that university calculus could make short shrift of many of the math problems that we were given. I decided to write a series of posts on these occurrences, which should give the average person an edge in high school. Some of these techniques borrow from college mathematics,  or simply got omitted from the Singapore-Cambridge syllabus for no good reason (AM-GM-HM inequality, L'Hôpital's rule, partial derivatives, differential operators etc.), hence I assume that the reader will at least make the effort to comprehend these, or already know these.

Part 1: Carrying the integral into the complex plane

One of the things that I pointed out was that we could integrate certain functions, and in particular,  trigonometric products, much faster than we were conventionally taught. One of my teachers discussed the legality of these 'tricks' with another, and as a result, I was responsible for the cheeky "Using integration by parts" printed ahead of a question on my final year examination.

For example, you could get up to 5 marks from "Find $\int e^{x}\cos2x\, dx$"

Oh, exploitable.

 $\int e^{x}\cos2x\, dx=\dfrac{e^{x}}{5}\left(\cos2x+2\sin2x\right)+c$

I didn't know the answer beforehand here: I did actually work it out in that single step. It took about 3 seconds for the answer to lay out visually in my head. The secret?

Some will notice that it can be shown that

$\int e^{ax}\sin bx\, dx=\dfrac{e^{ax}}{a^{2}+b^{2}}\left(a\sin bx-b\cos bx\right)+c$

$\int e^{ax}\cos bx\, dx=\dfrac{e^{ax}}{a^{2}+b^{2}}\left(a\cos bx+b\sin bx\right)+c$

which can be memorized. I'm not good with memorizing things, and moreover, I firmly believe that math should be used like a language (not true, I studied that there's independence on language faculties when I wrote my linguistics paper): we rarely put the effort to memorize our languages. But having integrated products of trigonometric functions quite a number of times, I find it very easy to remember these two identities. Let's illustrate what I mean with the first identity.

$\int e^{ax}\sin bx\, dx$
$=\int e^{ax} Im\left(e^{ibx}\right)\,dx$
$=Im\left(\int e^{ax}e^{ibx}\,dx\right)$
$=Im\left[e^{ax}\dfrac{1}{a+bi}\left(\cos bx+i\sin bx\right)\right]+c$

Here on, we only require some computational fortitude. Rewrite $\dfrac{1}{a+bi}$ to extract the real constant $\dfrac{1}{a^2+b^2}$; and what remains quite blatantly pins the imaginary part...

$Im\left[e^{ax}\dfrac{1}{a+bi}\left(\cos bx+i\sin bx\right)\right]+c$

$=\dfrac{e^{ax}}{a^{2}+b^{2}}\, Im\left[\left(a-bi\right)\left(\cos bx+i\sin bx\right)\right]+c$

$=\dfrac{e^{ax}}{a^{2}+b^{2}}\left(a\sin bx-b\cos bx\right)+c$

Of course, most of these steps can be done mentally, which is why it's very easy to remember the two identities, and this also makes it faster to carry the integral into the complex plane instead of integrating by parts.