November 12th, 2010, 05:37 AM  #1 
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  Integral Computation. [color=#000000]Compute the following integral: $\displaystyle \bf I=\int_{0}^{\frac{\pi}{2}}\frac{\cos(x)^{\sin(x)}} {\cos(x)^{\sin(x)}+\sin(x)^{\cos(x)}}\;dx$ .[/color] Last edited by skipjack; February 17th, 2018 at 11:19 PM. 
November 12th, 2010, 06:05 AM  #2 
Senior Member Joined: Dec 2009 Posts: 150 Thanks: 0  Re: Integral Computation. 
November 12th, 2010, 06:48 AM  #3 
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  Re: Integral Computation. I think that it makes it too comlicated . Try to see what happens from $\displaystyle \bf 0$ to $\displaystyle \bf\frac{\pi}{4}$ and from $\displaystyle \bf\frac{\pi}{4}$ to $\displaystyle \bf\frac{\pi}{2}$. See the plot below. Last edited by skipjack; February 18th, 2019 at 10:51 PM. 
November 12th, 2010, 04:18 PM  #4 
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  Re: Integral Computation.
$\displaystyle \bf I=\int_{0}^{\frac{\pi}{2}}\frac{\cos(x)^{\sin(x)}} {\cos(x)^{\sin(x)}+\sin(x)^{\cos(x)}}\;dx$ $\displaystyle =\int_{0}^{\frac{\pi}{4}}\frac{\cos(x)^{\sin(x)}}{ \cos(x)^{\sin(x)}+\sin(x)^{\cos(x)}}\; dx+\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos( x)^{\sin(x)}}{\cos(x)^{\sin(x)}+\sin(x)^{\cos(x)}} \;dx$ $\displaystyle =\int_{0}^{\frac{\pi}{4}}\frac{\cos(x)^{\sin(x)}}{ \cos(x)^{\sin(x)}+\sin(x)^{\cos(x)}}\; dx+\int_{\frac{\pi}{4}}^{0}\frac{\cos(\frac{\pi}{2 }t)^{\sin(\frac{\pi}{2}t)}}{\cos(\frac{\pi}{2}t)^{\sin(\frac{\pi}{2}t)}+\sin(\frac{\pi}{2}t)^{\cos(\frac{\pi}{2}t)}}\;d(\frac{\pi}{2}t)$ $\displaystyle =\int_{0}^{\frac{\pi}{4}}\frac{\cos(x)^{\sin(x)}}{ \cos(x)^{\sin(x)}+\sin(x)^{\cos(x)}}\; dx+\int^{\frac{\pi}{4}}_{0}\frac{\sin(t)^{\cos(t)} }{\cos(t)^{\sin(t)}+\sin(t)^{\cos(t)}}\;dt$ $\displaystyle =\int_{0}^{\frac{\pi}{4}}\frac{\cos(x)^{\sin(x)}}{ \cos(x)^{\sin(x)}+\sin(x)^{\cos(x)}}\; dx+\int^{\frac{\pi}{4}}_{0}\{1\frac{\cos(t)^{\sin(t)}}{\cos(t)^{\sin(t)}+\sin(t) ^{\cos(t)}}\}\;dt$ $\displaystyle \longrightarrow^{t=x}\int_{0}^{\frac{\pi}{4}}\frac {\cos(x)^{\sin(x)}}{\cos(x)^{\sin(x)}+\sin(x)^{\co s(x)}}\;dx+\int^{\frac{\pi}{4}}_{0}\{1\frac{\cos(x)^{\sin(x)}}{\cos(x)^{\sin(x)}+\sin(x) ^{\cos(x)}}\}\;dx$ $\displaystyle =\int_{0}^{\frac{\pi}{4}}\frac{\cos(x)^{\sin(x)}}{ \cos(x)^{\sin(x)}+\sin(x)^{\cos(x)}}\;dx\int^{\frac{\pi}{4}}_{0}\frac{\cos(x)^{\sin(x)}}{\ cos(x)^{\sin(x)}+\sin(x)^{\cos(x)}}\; dx+\int^{\frac{\pi}{4}}_{0}1\;dx$ $\displaystyle =\int_{0}^{\frac{\pi}{4}}1\;dx$ $\displaystyle =\frac{\pi}{4}$ Last edited by skipjack; February 17th, 2018 at 11:17 PM. 
November 12th, 2010, 04:45 PM  #5 
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  Re: Integral Computation. [color=#000000]Well done my friend! [/color] 
November 12th, 2010, 05:06 PM  #6 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: Integral Computation.
Very interesting problem! ...I entered that into my calculator earlier and aborted after about 15 minutes of it saying it was busy! 

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