
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
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 
Senior Member Joined: Aug 2014 From: United States Posts: 137 Thanks: 21 Math Focus: Learning  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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Integral of Exponential Function  Aqil  Calculus  2  October 29th, 2012 02:48 PM 
Exponential integral  abotaha  Calculus  10  July 28th, 2010 07:07 AM 
exponential and logarithm integral  Aurica  Calculus  2  June 10th, 2009 09:26 AM 
integral of exponential  alpacino  Calculus  1  March 2nd, 2009 06:06 AM 
integral of exponential  alpacino  Calculus  5  February 23rd, 2009 05:55 PM 