# looking for an explicit answer to my integral

4 visualizzazioni (ultimi 30 giorni)
Amir il 29 Feb 2012
Dear All,
In order to solve a physical problem, I need an explicit answer to the following integral: int(asin(1-1/x))
now I realize than an explicit answer might not exist and so that is perhaps why the 'int' function does not work. but I was wondering if you guys any suggestions for me as to how approach this problem and whether making approximations or assumptions would work?
Thanks, - Amir
##### 0 CommentiMostra -2 commenti meno recentiNascondi -2 commenti meno recenti

Accedi per commentare.

### Risposte (5)

Walter Roberson il 29 Feb 2012
int(arcsin(1-1/x),x=a..b) assuming a::real, b::real, a<=b
piecewise(a < b, piecewise(b < 0, arcsin((-1+b)/b)*b+(-1+2*b)^(1/2), b = 0, I, 0 < b, arcsin((-1+b)/b)*b-(-1+2*b)^(1/2))+piecewise(And(0 < b, a < 0), 2*I, 0)+piecewise(a < 0, -arcsin((-1+a)/a)*a-(2*a-1)^(1/2), a = 0, I, 0 < a, -arcsin((-1+a)/a)*a+(2*a-1)^(1/2)), b = a, 0)
In the above, "I" is sqrt(-1)
If your domain of integration is complex, then probably things get messy.
Decoding the piecewise: if the limits of integration are both on the same side of 0, then
b * arcsin(1-1/b) - a * arcsin(1 - 1/a) - (sqrt(2*b - 1) - sqrt(2*a-1))
and if they are on different sides of 0, then add 2*I to the above result.
If either of the limits of integration can be exactly 0 then the result gets more complicated.
##### 2 CommentiMostra NessunoNascondi Nessuno
Amir il 29 Feb 2012
Thx, the limits of my integral are 0+ and pi/2.
I am actually trying to solve this whole integral, below:
int(asin(1+c/sin(x)) + int(asin(1-c/sin(x))
where c is a constant
Walter Roberson il 29 Feb 2012
Your brackets do not match. You have more '(' than ')'
I do not seem to be able to locate any useful conversions for you for this more general case; sorry.

Accedi per commentare.

UJJWAL il 29 Feb 2012
The explicit closed form answer exists. It is following ,
a * (cos(1)*ci(1/x) + sin(1)*si(1/x)+x*sin(1)*cos(1/x)-x*cos(1)*sin(1/x))
Here , ci(x) means cosine Integral of x. si(x) means sine Integral of x.
which you can evaluate easily in MATLAB.
##### 2 CommentiMostra NessunoNascondi Nessuno
UJJWAL il 29 Feb 2012
cos(1) = 0.5403
sin(1) = 0.8415
Amir il 29 Feb 2012
thx, I am confused. What is a?

Accedi per commentare.

Jonathan Sullivan il 29 Feb 2012
x.*(asin((x-1)./x)-sqrt(2*x-1)./x^2))
##### 1 CommentoMostra -1 commenti meno recentiNascondi -1 commenti meno recenti
Amir il 29 Feb 2012
Thx Jonathan, could you explain your rationale? is that partial derivative technique?

Accedi per commentare.

Amir il 29 Feb 2012
Thanks guys. Could you guys explain how you arrived at these solutions though? I seem not to be able to follow! Thx!
##### 1 CommentoMostra -1 commenti meno recentiNascondi -1 commenti meno recenti
Amir il 29 Feb 2012
I am actually trying to solve this integral, below:
int(asin(1+c/sin(x)) + int(asin(1-c/sin(x))
where c is a constant

Accedi per commentare.

UJJWAL il 29 Feb 2012
Hi Amir ,
To solve the integral of a sin(1-1/x), follow the following steps,
a) Expand sin(1-1/x) and then integrate. b) You wil encounter the integral of sin(1/x) and cos(1/x). For solving cos(1/x) use the integration by parts by taking cos(1/x) as the first function and 1 as the second function. You will in process get expressions for Sin Integral, which is evaluated numerically.
In the same way you can solve for cos(1/x) also Finally rearrange the terms and you will get the answer I have given you . :-)
Hope this helps
Happy to help
UJJWAL
##### 2 CommentiMostra NessunoNascondi Nessuno
Amir il 29 Feb 2012
Hello Ujjwal,
Thanks for the help.
I see that you have considered a as a constant. I actually meant to ask for the integral of an arcsin, which in Matlab, is represented by asin. so 'a' is not a constant.
do you happen to know whether there is any solution to the following integral?
arcsin ( 1 - c/sin(x) ) where c is a constant.
UJJWAL il 29 Feb 2012
I am very sure that a closed form solution does not exist for this as there are bounds within which this integral would really make sure and those bounds depend on the values of x within which you are integrating.

Accedi per commentare.

### Categorie

Scopri di più su Programming in Help Center e File Exchange

### Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Translated by