Memorizing substitutions for HL Mathematics

[XeoKnight]

Soooo.... the other day, I remembered that while going through the syllabus we had been told that there would be a few substitutions we would be required to memorize for integration by substitution.

However, in the syllabus, we have this:

What exactly are 'standard substitutions' that we have to memorize? It's very vague to me.

These are the ones our teacher gave us:

integral of sqrt(a^2 - x^2) dx => x = asin(theta)

integral of sqrt(x^2-a^2) dx => x = asec(theta)

integral of sqrt(x^2+a^2) dx => x = atan(theta)

(unfortunately, I don't remember how to get theta...)

Are there any others we need to know? And I would be grateful if anyone could kick start my memory as to how to get theta

EDIT: Lemme rephrase, are there any useful substitutions that we can memorize? Such as these ones? These you can probably solve brute force, but in some questions just knowing these can be helpful. ctrls provided a few useful ones a bit down.

Edited May 4, 2014 by XeoKnight

[Maks123456]

Google the Mathematics formula booklet.

All the integrals provided are there.

[Maks123456]

Oh. I don't really know what you mean by getting theta, but if you have done a substitution with x = 2sin(theta),

arcsin(x/2)= theta

[XeoKnight]

Google the Mathematics formula booklet.

All the integrals provided are there.

I know you can do that... but what I'm talking about are substitutions not provided by the booklet. As you may have noticed, those three weren't on the booklet and were used in the specimen paper (I did it for my mock).

Another example of an integral you should memorize is the integral of tan(x) (which is ln(sec x) +c btw), but that one you may be asked to show using integration by parts. It's useful to memorize and not have to do at the time in the exam. These ones are others like that.

[Maks123456]

Are you sure you have been using a correct formula booklet? Because the one I have, which is for first examinations in 2014, it has the three integrals you have mentioned plus the tanx one.

[XeoKnight]

Are you sure you have been using a correct formula booklet? Because the one I have, which is for first examinations in 2014, it has the three integrals you have mentioned plus the tanx one.

Wait, really? Sorry then, I must have sounded pretty rude. I've been using the one provided to us beginning last year. I've just looked through the first page on Google for the booklets there, none of them have those substitutions, could you by chance link me yours if you have it online?

[ctrls]

A useful rule which includes tan x is .

I don't know if this is part of the syllabus, but reciprocal quadratics can be solved by completing the square. The derivatives page of the formula booklet is also worth keeping in mind, since the inverse of differentiation is integration. In particular the derivatives of sec/csc/cot are pretty hard to spot, but do come up from time to time.

For integration by parts, there is the LIATE rule. I don't personally use it, but I thought it may be worth mentioning. I guess it's also worth keeping in mind that logarithmic and inverse trig functions can be integrated this way, by setting v'=1.

Are you sure you have been using a correct formula booklet? Because the one I have, which is for first examinations in 2014, it has the three integrals you have mentioned plus the tanx one.

You may be confusing them with the inverse trig ones, but the three he listed aren't in the formula booklet. The integrals he's referring to are,

Edited May 3, 2014 by ctrls

[Maks123456]

http://ibmaths.pbworks.com/w/file/fetch/73875983/HL%20Formula%20Booklet%20%282014%29.pdf

But integral of tanx is not there, sorry. My bad.

ctrls, the integral you have given can be solved using a trig substitution, so I do not understand what you are implying.

[XeoKnight]

Err, in which case none of my examples are on the formula booklet lol, if you can solve it using a trig substitution.

Lemme rephrase it then, are there any useful substitutions that we can memorize? Like the ones that me and ctrls mentioned?

[Fiz]

I guess you should memorize them, or atleast I did. You can also check the specimen in section B paper1, there is a question that is based on such techniques.

EDIT: You might also want to check the markscheme for that question to find other alternative methods if you are not comfortable with this one.

Edited May 4, 2014 by Fiz

[Rigel]

Hello. Basically the only substitutions (and techniques) you need to know are:

- All types of trig substitutions (no need to know the hyperbolic ones, as you can derive them as you go). If you don't use the triangle when making a substitution of the type x = sin(t), you can always rewrite as t = arcsin(x) and then substitute accordingly (i.e. finding sin(arctanx)).

- Substituting in the case f(x) and f'(x) (seeing the differential in a part of the equation).

- The special case to obtain the integrals for sec(x) and csc(x) (kind of a hidden substitution you need to memorize).

- Integration by parts.

- Partial fractions. A special note is an integral of the type say (2x-4)/(x²-3x+4). As you see, the differential of the denominator is 2x-3. Now, a nice trick is to rewrite the partial fraction as follows: (2x-4+1-1)/(x²-3x+4) = (2x-3)/(x²-3x+4) + (-1)/(x²-3x+4), in which you can solve by completing the square and making the appropriate substitutions.

The rest can be deduced/obtained easily. A trick I always use for trig substitution is comparing the expressions with trig identities. If we have an expression of the type sqrt(x²-4), a substitution that comes to mind is x = 2sec(t) and then rewrite in terms of tan(t).