looking for an explicit answer to my integral
1 visualizzazione (ultimi 30 giorni)
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
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.
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.
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