Hi!
Thank you for the reply! Yes I am talking about a change in variables not limits - sorry that's my fault I didn't think of the notation I was using.
HallsofIvy - thank you! I thought that was the case but was confused how to so it. Basically I've got a function g(s+) and g(s-) and a...
Homework Statement
I am performing a change of variables, s --> t and am wondering can I just write g(s) --> g(t) or do I have to alter the function e.g. g(s) --> a*g(t).
Homework Equations
s = (a/b)* t
g(s) is to be found numerically and therefore we do not have it's definition.
So...
Once you complete the square, substitute x-1/2 = 1/2sint to get 1/4 - 1/4sin^2 t
Factorise to get 1/4 ( 1-sin^2 t) = 1/4 cos^2 t
As we are looking at sqrt (x-x^2), take sqrt of this to get 1/2 cost
Does that make sense?
Oh okay, I'm a bit lost. I thought you could introduce the dummy variable and then differentiate/integrate as normal as you changed limits etc. Could you explain how it would be different?
Thanks
To work out v i:
1. Increased the power of the bracket from -3 to -2.
2. Divide by the new power (-2)
3. Divide by the differential of the bracket (2t)
Could you explain where I went wrong? Thanks Harriet
That is a great help I think!
I have worked through what you did, and got:
u=t^2
du = 2t dt
dv = (1+t^2)^-3
v = (1+t^2)^-2 / -4t
so integral of udv =
((t^2)(1+t^2)^-2) / (-4t) evaluated 0-1
+ integral_0 ^1 (1+t^2)^(-2) / -2 dt (the -4t from v has canceled with the 2t...
Homework Statement
Integrate (x-x^2)^1/2 from 1/2 to 1.
I tried to use substitution with x=sinu, dx=cos du to get:
sinu(1-sinu)^1/2 cosu du
but no idea where to go from there!
Also tried integration by parts, but it didn't work!
Help!
NB I tried to write this as Latex but not sure how to transfer it! If you can give me a hint I will try to repost so that it is easier to read!
I think so...
This has singularies at s=0 and s=1, same as my example.
Start with $s = e^{iz}$ and so $ds = ize^{iz} dz$
I've left it in terms...
Homework Statement
Show that the formula:
∫_0^∞ (s^a-1)/(s-1) ds = -pi cot (a pi)
may be calculated by considering an analytical branch of function:
f(z) = z^(a-1) / (z-1)
and integrating along a contour consisting of:
a) a circle radius R, centred at (0,0)
b) with a branch cut running...