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Let the integral be I. Now differentiate it with respect to a and after solving some more steps, you arrive at a differential equation in I and a i.e.,
dI/da + I/a = b*ln(sin(ab)).
Solve this differential equation to get I.
P.S.- Sorry, I dunno latex....else I would've posted my solution.
It is a bit surprising that this can be integrated at all, since the function ln(sin(x)) becomes infinite as x -> 0. (But this can be proven by considering the integral of ln(x) from 0 to b.)
Although I doubt that the function ln(sin(ax)) can be integrated explicitly in terms of "elementary" functions — and we better assume that a and b are both positive here ((or else both negative. But we will assume they are both positive).
Now assume a = 1. We want the function ln(sin(x)) to be real, and for this we need to avoid values of sin(x) that are less than 0. So we must limit the values of x near 0 to 0 <= x <= pi.
So let us set F(x) := the integral from 0 to x of ln(sin(t)) dt.
We know that F(0) = 0. By using the advanced complex variables technique of contour integration (in an advanced way), it can be determined that F(pi/2) = -pi ln(2)/2 and, by the symmetry of sin(x), that also F(pi) = -pi ln(2).
If we like, we can extend this function to all x via F(x) := the integral from 0 to x of ln|sin(t)| dt.
Then the graph of y = F(x) is a beautiful wavy curve that wiggles about the line y = -ln(2) x, and takes the exact values of F(N pi/2) = -N pi ln(2)/2 for all integers N.
Extending these facts to values of a other than a = 1 is an easy task.
@Tanishq Varshney I don't think that you are active nowadays............but if this helps................This paper can maybe help you out.................!!!
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This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
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@Tanishq Varshney I don't think so this integration is possible with normal methods. Try substituting: sinax=2ieiax−e−iax
@Kartik Sharma @Isaac Buckley @Ishan Dasgupta Samarendra @Satyajit Mohanty
Let the integral be I. Now differentiate it with respect to a and after solving some more steps, you arrive at a differential equation in I and a i.e.,
dI/da + I/a = b*ln(sin(ab)).
Solve this differential equation to get I.
P.S.- Sorry, I dunno latex....else I would've posted my solution.
It is a bit surprising that this can be integrated at all, since the function ln(sin(x)) becomes infinite as x -> 0. (But this can be proven by considering the integral of ln(x) from 0 to b.)
Although I doubt that the function ln(sin(ax)) can be integrated explicitly in terms of "elementary" functions — and we better assume that a and b are both positive here ((or else both negative. But we will assume they are both positive).
Now assume a = 1. We want the function ln(sin(x)) to be real, and for this we need to avoid values of sin(x) that are less than 0. So we must limit the values of x near 0 to 0 <= x <= pi.
So let us set F(x) := the integral from 0 to x of ln(sin(t)) dt.
We know that F(0) = 0. By using the advanced complex variables technique of contour integration (in an advanced way), it can be determined that F(pi/2) = -pi ln(2)/2 and, by the symmetry of sin(x), that also F(pi) = -pi ln(2).
If we like, we can extend this function to all x via F(x) := the integral from 0 to x of ln|sin(t)| dt.
Then the graph of y = F(x) is a beautiful wavy curve that wiggles about the line y = -ln(2) x, and takes the exact values of F(N pi/2) = -N pi ln(2)/2 for all integers N.
Extending these facts to values of a other than a = 1 is an easy task.
I can integrate that, if a=1 or 2, and b=2nπ. In that case, the answer to the integral would be =−2nπln(2).
I don't think I can integrate it for other values of a,b=0.
using complex numbers , its easy.
@Tanishq Varshney I don't think that you are active nowadays............but if this helps................This paper can maybe help you out.................!!!
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@Satyajit Mohanty @Ishan Dasgupta Samarendra