looking for an explicit answer to my integral

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Amir
Amir 2012-2-29
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

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回答（5 个）

Walter Roberson
Walter Roberson 2012-2-29
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.
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Amir
Amir 2012-2-29
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
Walter Roberson 2012-2-29
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.

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UJJWAL
UJJWAL 2012-2-29
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.
Jonathan Sullivan
Jonathan Sullivan 2012-2-29
x.*(asin((x-1)./x)-sqrt(2*x-1)./x^2))
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Amir
Amir 2012-2-29
Thx Jonathan, could you explain your rationale? is that partial derivative technique?

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Amir
Amir 2012-2-29
Thanks guys. Could you guys explain how you arrived at these solutions though? I seem not to be able to follow! Thx!
  1 个评论
Amir
Amir 2012-2-29
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

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UJJWAL
UJJWAL 2012-2-29
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 个评论
Amir
Amir 2012-2-29
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
UJJWAL 2012-2-29
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.

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