Integration and reverse chain rule (1 Viewer)

[nottellingu]

I was wondering how much working out we need to show when using the reverse chain rule. For example when integrating sin^4x cos x dx can we just claim than I= sin 5x/5 + c. Since I is in the form f'(x){f(x)}^n therefore it is = f(x) ^n+1/(n+1) +c.

I am aware that this q can also be done by substituition but the reverse chain rule saves alot of time !

 

[YannY]

in the hsc that one step is okay, unless you want to be absolutely sure you are getting the mark then use a substitution method =] in this case substitute u=sinx.

For school assesments: ask your teacher.

 

[nottellingu]

If its a straight forward question ill probably do it by substituition but the reverse chain rule is so much easier especially when u get an expression in the standard form, especially after a lengthy question e.g. after using the t-formula.

 

[YannY]

haha thats true and you'll get the mark in the hsc doing that, sometimes in school, teachers are idiots, e.g one of my teacher said you cant prove 3 points are collinear by finding the gradient of them......

Yeah it is faster but it's easier to make a mistake using that reverse chain rule

Better safe than sorry.

 

[Trebla]

As long as you get the answer correct you'll get full marks even without working. If it's obvious what the answer is then there's no point in making a substitution and wasting time. You're more likely to do silly things like forgetting to resubstitute back to the original variable by doing a substitution on an obvious one like that.

Finding the primitive is really nothing more than guessing what function you need to differentiate to get the expression being integrated...

Anything that takes the form f'(g(x))g'(x) can be indefinitely integrated to f(g(x)) + c easily without substitution. Substitution is essentially let u = g(x) => du = g'(x)dx, so integral of f'(u) du = f(u) + c

Substitution is really only useful if g(x) is a very complicated function and it is not obvious what the primitive function is.

 

[Jinpoo]

Better safe than sorry.

i beg to differ. Better fast than slow is the more precise saying!

why would you waste a minute or two doing useless steps of substitution when you can just 3-line it!

 

[Trajan]

i beg to differ. Better fast than slow is the more precise saying!

why would you waste a minute or two doing useless steps of substitution when you can just 3-line it!

You could make a costly error Jin-poop,that's why.

In the heat of the exam mistakes are likely to happen, double checking never hurt anyone.

 

[Trebla]

You could make a costly error Jin-poop,that's why.

In the heat of the exam mistakes are likely to happen, double checking never hurt anyone.

If the integral is simple like ∫xe^x²dx, you're more likely to make an error via substitution under the pressure of exam conditions. The most common mistakes in substitutions are forgetting to substitute back to the original variable and forgetting to change the integration bounds for definite integrals. You can avoid these very easily by simply stating the correct primitive in one line using 'reverse chain rule'.

You should really resort to a substitution if the question requires you to do so, or if the integral is more complicated. However, some people actually have difficulty applying the reverse chain rule mentally and just go for substitution anyway. If that's the case with you then by all means go for a substitution, otherwise if you're mentally capable of visualising the integrand as a manipulated form of a function of a function derivative, then use the reverse chain rule (which is advantageous if you can do it).

Also, given the limited time in the Extension 2 examination, it is highly unlikely that you will be able to double check your working without sacrificing marks in a non-attempt of other questions. You have to get it right the first time!

 

[shinn]

I agree with trebla, using u = blah blah substitution and then subbing back takes too long. Imo, writing more lines doesn't always prove to be efficient.

 

[YannY]

As you know, i totally agree with you. But this is not dependent on your opinion. Teachers become teachers because they never got a good uai and many teachers dont know what they are doing, sometimes without writing enough they would randomly blame you for doing the wrong thing thinking they are helping you. In the end its up to you, one extra step is easier to check and helps reduce mental stress. Its up to you in the end, if you go to a private school lucky you, i dont so thank you good nite.

 

[Affinity]

as long as you know what you are on about they will give you the marks...

math teachers at state schools were quite good, atleast when I did the HSC

 

[Slidey]

But this is not dependent on your opinion. Teachers become teachers because they never got a good uai and many teachers dont know what they are doing, sometimes without writing enough they would randomly blame you for doing the wrong thing thinking they are helping you.

That's fairly rubbish, man.

 

[shinn]

As you know, i totally agree with you. But this is not dependent on your opinion. Teachers become teachers because they never got a good uai and many teachers dont know what they are doing, sometimes without writing enough they would randomly blame you for doing the wrong thing thinking they are helping you. In the end its up to you, one extra step is easier to check and helps reduce mental stress. Its up to you in the end, if you go to a private school lucky you, i dont so thank you good nite.

Not true.... I know some one from my school who got 100 uai and decided to go for a teaching degree.