random thread

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With random questions

- How do you do ? nvm figured it out


Which can be solved using substitution


Also, is it possible to use the weierstrass substitution to brute force any trig integral which can also be solved using other techniques? Or is it only useful for certain trig integrals?

thx

[Ilovemathsmeth]

Random: that looks beautiful, I've seen the substitution method of integration and it looks amazing.

~crazyyy about integration~

[zzdfa]

I'm not familiar with the weierstrass substition, but this works for any rational function of trig funtions:

http://en.wikipedia.org/wiki/Integration_using_Euler's_formula

[/0]

I'm not familiar with the weierstrass substition, but this works for any rational function of trig funtions:

http://en.wikipedia.org/wiki/Integration_using_Euler's_formula

thanks zzdfa
I have heard of complex integration but i didn't know it is that powerful

[Ahmad]

You can try them both on this, compare and contrast,



there's also a really clever and well known trick for this one, that I wasn't clever enough to discover on my own but you might want to try,

here's a hint: let and , consider both bJ + aI and aJ - bI.

[/0]

Thanks! Let's see what happens...

With weierstrass ,     , , can be derived.



With tedious partial fractions,



With ,







(like I've run a marathon)



With the Ahmad's trick:











(Much better!!!!!)


With complex numbers, ,





....aand I'm stuck

It's that stupid in the denominator, I'm not sure how to get rid of the i.

[Ahmad]

Well done, glad you liked it

When you get stuck with exponentials like that and you can't think of anything better you can always try the substitution to get an integral of the ratio of two polynomials which has been solved in theory.

[kenhung123]

Is that geometry in 4 dimensions?

[/0]

edit: lol you meant my avatar?

Yeah it's a hypercube rotating in 4 dimensional space. What you are seeing is the 3-dimensional 'shadow' of the hypercube


Not really, it's just practicing integration techniques.

You'll come across trig integration in methods and if you do spesh there are some more advanced techniques. I'm currently using Stewart Calculus, which Damo kindly uploaded http://vcenotes.com/forum/index.php/topic,20178.0.html

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What's the difference between the line integrals



and



Where is the tangent unit vector

In stewart calculus they are both "line integrals over C"

[zzdfa]

the first one, you're integrating over a scalar function. e.g. finding the total mass of a wire given that the mass density at any point x,y is the scalar f(x,y).

the second one, over a vector field. e.g. finding  work done moving along a curve C given that the field at any point x,y is the vector F(x,y).

[/0]

Ah ok, so the direction of the path matters in the second one but not the first one?

Otherwise are they pretty much the same?

[zzdfa]

hmm. Do you know what scalar fields and vector fields are?

lets imagine we're on R^2, the plane. imagine it's been split into tiny grid squares.

in the first case, every square is assigned a scalar value (so this is a scalar field) and when we integrate over the path C what happens is we walk along the path and add up the values of all the points we pass through.


the 2nd case is when every square is assigned a vector (so this is a  vector field). we walk along the path C and for each square we pass through, we work out the component of the vector that is in the direction we are travelling (hence the F dot T), and add the magnitude of that to our score.

for example, if you want to cross a river, the effort you expend shouldn't depend on how fast the water is flowing downstream, because it's pushing you perpendicular to the direction you're heading (dot product = 0 ).

[/0]

Ah ok, thanks very much zzdfa

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Let

In wikipedia it says if , evaluates to

However, going through the derivation,





Why is it ok to get rid of the modulus?