Do I need to memorize certain integration facts?

http://dl.dropbox.com/u/3579325/Untitled.jpg

Hi, I am currently learning integration.

I encountered an integration question in the top left of the attached image, ya solution provided.

My teacher said that I needed to memorise all 3 formulas at the bottom right of image to solve the question and similar questions in future.

Are they basic formulas that I must memorise?

If not, can someone show me how to derive on the spot?

I believe Maths should be testing LOGIC(Thinking) and APPLICATION(Wondering), not on MEMORY WORK(Nerd) even though some formulas have to be memorise like those 3 on the bottom left.

Any advice?

Thanks.(Bow)

re: Do I need to memorize certain integration facts?

Quote:

Originally Posted by

**normalguy** http://dl.dropbox.com/u/3579325/Untitled.jpg
Hi, I am currently learning integration.

I encountered an integration question in the top left of the attached image, ya solution provided.

My teacher said that I needed to memorise all 3 formulas at the bottom right of image to solve the question and similar questions in future.

Are they basic formulas that I must memorise?

If not, can someone show me how to derive on the spot?

I believe Maths should be testing LOGIC(Thinking) and APPLICATION(Wondering), not on MEMORY WORK(Nerd) even though some formulas have to be memorise like those 3 on the bottom left.

Any advice?

Thanks.(Bow)

Every subject, including calculus, has a grammar, a dialectic, and a rhetoric. The grammar consists of the basic facts - the nuts and bolts of the subject. The dialectic is the logic: how do those nuts and bolts fit together? The rhetoric is the persuasive element: now that you have the truth (grammar plus dialectic), how do you persuade others of that truth?

I am definitely a fan of memorizing the grammar of a subject. For calculus, what would that be? Basic derivative and integral facts. Of the formulas you have written there, you definitely need to know your sine and cosine. I would say all the others are optional. You can derive them all (pun intended) by knowing how to differentiate all the basic trig functions: just turn each differentiation rule into an integral rule.

That's how I would proceed. Make sense?

Re: Do I need to memorize certain integration facts?

Do you mean memorise or not?

I know how to differentiate -cotx, secx and -cosecx.

Without memorising the integral of cosec^2 x, secxtanx and cosecxcotx, how do you integrate them?

Are there anyway to derive the integrals other than randomly picking basic trig functions to differentiate till the derivative match the function to be integrated?

Re: Do I need to memorize certain integration facts?

Quote:

Originally Posted by

**normalguy** Do you mean memorise or not?

I know how to differentiate -cotx, secx and -cosecx.

Without memorising the integral of cosec^2 x, secxtanx and cosecxcotx, how do you integrate them?

Are there anyway to derive the integrals other than randomly picking basic trig functions to differentiate till the derivative match the function to be integrated?

I agree with the OP - I personally only remember a few basic derivatives and integrals. The way I would integrate that function is first to get it all to trigonometric functions, then to sines and cosines.

$\displaystyle \displaystyle \begin{align*} \int{\frac{\csc{(\ln{\theta})}\cot{(\ln{\theta})}} {\theta}\,d\theta} &= \int{\csc{u}\cot{u}\,du}\textrm{ after making the substitution }u=\ln{\theta} \implies du = \frac{d\theta}{\theta} \\ &= \int{\frac{\cos{u}}{\sin^2{u}}\,du} \\ &= \int{\frac{1}{v^2}\,dv}\textrm{ after making the substitution }v = \sin{u} \implies dv = \cos{u}\,du \\ &= \int{v^{-2}\,dv} \\ &= \frac{v^{-1}}{-1} + C \\ &= -\frac{1}{v} + C \\ &= -\frac{1}{\sin{u}} + C \\ &= -\csc{u} + C \\ &= -\csc{(\ln{\theta})} + C \end{align*}$

Re: Do I need to memorize certain integration facts?

(Clapping)then, how would you integrate cosec^2x and secxtanx? Do you use substitution or by parts? Clueless

Re: Do I need to memorize certain integration facts?

well this is why we learn to differentiate first.

because then, you might perhaps recognize that sec(x)tan(x) is the derivative of sec(x), and that -csc^2(x) is the derivative of cot(x),

which makes integrating a bit easier.

Re: Do I need to memorize certain integration facts?

Quote:

Originally Posted by

**normalguy** (Clapping)then, how would you integrate cosec^2x

[snip]

You have either memorised what the integral is, or you have memorised that the derivative of cot(x) is -cosec^2(x), or you just have to use the Weiertstrass substitution t = tan(x/2) (which is the reason why it is good to memorise some things).

Quote:

Originally Posted by

**normalguy** (Clapping)then, how would you integrate

[snip]

secxtanx? Do you use substitution or by parts? Clueless

By re-writing it as $\displaystyle \frac{\sin(x)}{\cos^2(x)}$ and making the substitution u = cos(x).

Re: Do I need to memorize certain integration facts?

Ewwwwwwwwww this thread is my nightmare! **Substitution everywhere!** Yuck!

I understand what everyone is saying. However, I would learn all 6 basic elementary integrals and derivatives, personally. In my exam board, we are actually given almost all of them in a formula booklet **during** the exam, but having them at your fingertips can be a blessing. Yes, conversion to sin and cos will usually suffice, but it can often be long and tedious process and if you can just learn these twelve things (derivatives and integrals) then you just make it easier for yourself in the longterm. I know...it's all derived from substitution, but when you're being timed under exam conditions, you might not have time to go through rederiving every single integral you come across.

Of course, there are some shorthand rules that help. With all due respect, Mr F annoyed me **hugely** just now when he said:

"By re-writing it as $\displaystyle \frac{\sin(x)}{\cos^2(x)}$ and making the substitution $\displaystyle u = cos(x)$'" because it just immediately jumped out to me as what my teacher calls a 'related integral' which is one whereby the rule:

$\displaystyle \int f'(x)[f(x)]^n dx=\frac{[f(x)]^{n+1}}{n+1}+C$ applies perfectly. Maybe you haven't come across this rule yet? If not, you will do soon - and I won't challenge you to see how much time it saves when compared with the suggested substitution! The same method can be applied to Prove It's example, although not initially, again saving a lot of effort.

My point: personally, I do think it's worth learning them. It isn't essential, but it does give a slight edge, I believe.

Re: Do I need to memorize certain integration facts?

This is my country's A-level formula list:

http://www.seab.gov.sg/aLevel/2012Syllabus/ListMF15.pdf

Apparently,all 6 are not given except for integral of secxtanx. Why this not the rest?(Wondering)

For now, I planned to differentiate cosecx and cotx before I start doing the paper because I do not like the idea of memorising the integrals.

I would only memorise the derivative of sinx, cosx and tanx.

I am not sure if I should also memorise R-formula.

Anyway, how do you use the related integral method?

Thanks to all for participating so actively.