looking for an explicit answer to my integral

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

Answer 1

[Walter Roberson]

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.

[Amir]

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]

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.

Answer 2

[UJJWAL]

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.

[UJJWAL]

cos(1) = 0.5403

sin(1) = 0.8415

[Amir]

thx, I am confused. What is a?

Answer 3

[Jonathan Sullivan]

x.*(asin((x-1)./x)-sqrt(2*x-1)./x^2))

[Amir]

Thx Jonathan, could you explain your rationale? is that partial derivative technique?

Answer 4

[Amir]

Thanks guys. Could you guys explain how you arrived at these solutions though? I seem not to be able to follow! Thx!

[Amir]

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

Answer 5

[UJJWAL]

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

[Amir]

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]

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.