The antiderivative is Si(x), which is not an finite elementary antiderivative.
This function is well known and does not have an elemetry antiderivative.
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What is the exact wording of your question. If you need to find its derivative the Fundamental theorem of calculus will do the job!
This the very last part of a long, multi-part problem. I am trying to find the first positive inflection point to Si(x). I took the second derivative of Si(x), set this equal to zero, and used Newton's method to find the x-coordinate of the inflection point (which was 4.4934). All I have left to do is to find the y coordinate. I can do this easily with a graphing calculator or Mathematica, but I think that we have to do it the long way.
Is there an easier approach?
Thanks.
Since it does not have an elemetry antiderivative you a few options.
First you could use a quadrature rule to estimate the value of the integral
Simpsons rule, Trapezoid, midpoint or evena Riemann sum.
2nd an the more commin if you know infinite series
You can integrate this term by term to get an infinite series representation of the function and use the Alternating Series test to bounded the error as small as you wish.