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 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. This may help: http://mathnow.wordpress.com/2009/11/13 ... stitution/
November 12th, 2010, 06:48 AM   #3
Math Team

Joined: Nov 2010
From: Greece, Thessaloniki

Posts: 1,990
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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.
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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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