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 June 22nd, 2015, 09:05 PM #1 Senior Member   Joined: Aug 2014 From: United States Posts: 137 Thanks: 21 Math Focus: Learning integral of exponential of sin squared Compute the integral $\displaystyle\int_0^{\frac\pi 2} e^{\sin^2 x}dx$. When I tried, I made a careless (and wrong) assumption and got $\displaystyle\frac{\pi} {2}\sqrt e$. However, Wolfram Alpha gives my answer multiplied to the constant $I_0(1/2)$ where $I_n(z)$ is apparently a "Bessel function of the first kind." I am curious if anyone can show me how to derive this result. June 24th, 2015, 04:55 AM #2 Math Team   Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus Substitute $\sin(x)^2$ with $\frac{1-\cos(2x)}{2}$. June 26th, 2015, 07:37 AM   #3
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Joined: Aug 2014
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Quote:
 Originally Posted by ZardoZ Substitute $\sin(x)^2$ with $\frac{1-\cos(2x)}{2}$.
I have been at it for a while, but I don't see the connection to the Bessel differential equation.

EDIT: I may have got the Bessel differential equation wrong in the first place; I thought that $I_0(x)$ was the solution to $xy'' + y' +xy=0$ (although I don't know the initial conditions). Tags exponential, integral, sin, squared Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Aqil Calculus 2 October 29th, 2012 02:48 PM abotaha Calculus 10 July 28th, 2010 07:07 AM Aurica Calculus 2 June 10th, 2009 09:26 AM alpacino Calculus 1 March 2nd, 2009 06:06 AM alpacino Calculus 5 February 23rd, 2009 05:55 PM

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