#2 27. 10. 2015 10:57
Re: Trigonometric Integration.
, 
and that 
is an odd function on 
.![$
\int_{\frac{\pi}{2}}^{\frac{5\pi}{2}}\frac{\mathrm e^{\tan^{-1}(\sin x)}}{\mathrm e^{\tan^{-1}(\sin x)}+\mathrm e^{\tan^{-1}(\cos x)}}\,\mathrm dx
&=
\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}}\frac{\mathrm e^{\tan^{-1}(\sin x)}}{\mathrm e^{\tan^{-1}(\sin x)}+\mathrm e^{\tan^{-1}(\cos x)}}\,\mathrm dx
+
\int_{\frac{3\pi}{2}}^{\frac{5\pi}{2}}\frac{\mathrm e^{\tan^{-1}(\sin x)}}{\mathrm e^{\tan^{-1}(\sin x)}+\mathrm e^{\tan^{-1}(\cos x)}}\,\mathrm dx\\[.5\baselineskip]
&=
\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}}\frac{\mathrm e^{\tan^{-1}(\sin x)}}{\mathrm e^{\tan^{-1}(\sin x)}+\mathrm e^{\tan^{-1}(\cos x)}}\,\mathrm dx
+
\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}}\frac{\mathrm e^{\tan^{-1}(-\sin x)}}{\mathrm e^{\tan^{-1}(-\sin x)}+\mathrm e^{\tan^{-1}(-\cos x)}}\,\mathrm dx\\[.5\baselineskip]
&=
\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}}\frac{\mathrm e^{\tan^{-1}(\sin x)}}{\mathrm e^{\tan^{-1}(\sin x)}+\mathrm e^{\tan^{-1}(\cos x)}}\,\mathrm dx
+
\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}}\frac{\mathrm e^{-\tan^{-1}(\sin x)}}{\mathrm e^{-\tan^{-1}(\sin x)}+\mathrm e^{-\tan^{-1}(\cos x)}}\,\mathrm dx\\[.5\baselineskip]
&=
\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}}\frac{\mathrm e^{\tan^{-1}(\sin x)}}{\mathrm e^{\tan^{-1}(\sin x)}+\mathrm e^{\tan^{-1}(\cos x)}}\,\mathrm dx
+
\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}}\frac{\mathrm e^{\tan^{-1}(\cos x)}}{\mathrm e^{\tan^{-1}(\sin x)}+\mathrm e^{\tan^{-1}(\cos x)}}\,\mathrm dx\\[.5\baselineskip]
&=
\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}}\frac{\mathrm e^{\tan^{-1}(\sin x)}+\mathrm e^{\tan^{-1}(\cos x)}}{\mathrm e^{\tan^{-1}(\sin x)}+\mathrm e^{\tan^{-1}(\cos x)}}\,\mathrm dx
=
\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}}1\,\mathrm dx={\color{blue}\boldsymbol{\pi}}
$](/mathtex/b1/b1ebf07843bb1354c028141ca6ab7769.gif "kopírovat do textarea")
Backslash je v TeXu tak důležitý jako nekonečno při dělení nulou v tělesech charakteristiky 0.
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#3 05. 11. 2015 15:00
- stuart clark
- Příspěvky: 1015
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Re: Trigonometric Integration.
Thanks ↑ Pavel: Got it.
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