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November 12th, 2010, 05:37 AM   #1
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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.
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November 12th, 2010, 06:05 AM   #2
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Re: Integral Computation.

This may help:

http://mathnow.wordpress.com/2009/11/13 ... stitution/
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November 12th, 2010, 06:48 AM   #3
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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.
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November 12th, 2010, 04:18 PM   #4
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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.
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November 12th, 2010, 04:45 PM   #5
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Re: Integral Computation.

[color=#000000]Well done my friend! [/color]
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November 12th, 2010, 05:06 PM   #6
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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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