FP2 ArcSin method!

Stuck with something which comes up in the FP2 paper......

I don't understand why they're using this new formula with arcsin rather than the normal way of working this out....
....and obviously if you continue with the normal way you don't end up with Pi in the equation at all....

What am I doing wrong?

Edit: There should be a ^0.5 on after the first bracket on what I wrote above

(edited 13 years ago)

Reply 1

nuodai

17

There are lots of things wrong with your working... if you explain how you got from

(3-4x)^{-\frac{1}{2}}

to

\dfrac{3-4x^2}{-1/2} \left( 3x-\dfrac{4x^3}{3} \right)

then I'll let you know where you're going wrong.

A few incorrect things I can already see are:
- You've integrated term-by-term inside a bracket with a power. This is wrong: the integral of (1+x)² doesn't have x+x²/2 anywhere in it, for example.
- You've divided by... something, and definitely not the right thing
- You haven't used the chain rule when you need to

It looks like you're trying to say that

\displaystyle \int [f(x)]^n\, dx = \dfrac{1}{f'(x)} \times \dfrac{f(x)^{n+1}}{n+1}

or something, but this result isn't right (and even if it was you wouldn't have executed it correctly either).

Reply 2

matthewdownloads

OP

1

Using chain rule:

The equation is divided by -1/2 since this is the original power. Then +1 is added to the power.

Then this result is multiplied by the inside of the bracket integrated.

What's wrong with that?

Like I said I missed the power by mistake.

Reply 3

nuodai

17

Original post by matthewdownloads

What's wrong with that?

The chain rule states that

\dfrac{d}{dx}[f(g(x))] = f'(g(x))g'(x)

, and integration by substitution states that

\displaystyle \int f'(g(x))g'(x)\, dx = f(g(x))+C

. It doesn't state, for instance, that

\displaystyle \int f'(g(x))\, dx = \dfrac{f(g(x))}{g'(x)} + C

, but even this doesn't seem to follow what you've done. It looks like you're sort-of differentiating and sort-of integrating, and everything's gone horribly wrong.

I'll go through what you said step-by-step:

The equation is divided by -1/2 since this is the original power. Then +1 is added to the power.

When integrating something in the form

(ax+b)^n

you raise the power first and then divide by the new power (and also divide by the coefficient of x); dividing by the original power is what you do when you differentiate. However this only works for functions of the form

(ax+b)^n

, whereas here you have an

x^2

term in the bracket and so this won't work. You have to use a substitution instead (in this case,

2x=\sqrt{3}\sin u

works nicely).

Then this result is multiplied by the inside of the bracket integrated.

Why do you integrate what's inside the bracket? I have no idea where you've picked up this method from but it's completely wrong.

I mean, try differentiating what you get when you do this (before you put in the limits). You certainly won't get your original function.

Reply 4

dknt

12

Original post by matthewdownloads

Stuck with something which comes up in the FP2 paper......

I don't understand why they're using this new formula with arcsin rather than the normal way of working this out....
....and obviously if you continue with the normal way you don't end up with Pi in the equation at all....

What am I doing wrong?

Edit: There should be a ^0.5 on after the first bracket on what I wrote above

You've done some funky (and incorrect) integration. I assume by the normal way you mean by inspection.

If you consider

\dfrac{d}{dx}[(3-4x^2)^{\frac{1}{2}}]

We get

-4x(3-4x^2)^{-\frac{1}{2}}

. This almost what we need, except there is an extra -4x term, so this "normal way" (if that's what you mean), won't work hence the use of arcsin in the solution. You seemed to have done something strange with the chain rule backwards...

Reply 5

matthewdownloads

OP

1

Original post by dknt

You've done some funky (and incorrect) integration. I assume by the normal way you mean by inspection.

If you consider

\dfrac{d}{dx}[(3-4x^2)^{\frac{1}{2}}]

We get

-4x(3-4x^2)^{-\frac{1}{2}}

. This almost what we need, except there is an extra -4x term, so this "normal way" (if that's what you mean), won't work hence the use of arcsin in the solution. You seemed to have done something strange with the chain rule backwards...

Aha, I think its the classic error of me confusing my integration and differentiation. Basically I was treating it as though I was differentiating it, but using the integration rules of increasing powers/dividing etc. I think that's what's wrong.

Reply 6

nuodai

17

Original post by matthewdownloads

Aha, I think its the classic error of me confusing my integration and differentiation. Basically I was treating it as though I was differentiating it, but using the integration rules of increasing powers/dividing etc. I think that's what's wrong.

Something like that. It just looked like you were applying all sorts of differentiation and integration results more or less at random to different parts of the function you were presented with.

FP2 ArcSin method!

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you