User:Dan Nessett/Sandboxes/Sandbox 2: Difference between revisions

Revision as of 10:32, 17 September 2009

<section begin=chapter1 />this is a chapter<section end=chapter1 />

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-I think it is useful to get on record the following derivation. It doesn't belong in the proof itself, but putting it on this talk page gets it into the history of the article:
+<section begin=chapter1 />this is a chapter<section end=chapter1 />
-<math>
-x=\cos \theta\; \Longrightarrow\; dx=-\sin \theta d\theta\quad\hbox{and}\quad 1-x^{2} =(\sin \theta)^2 \Longrightarrow\; x^{2}-1 =\ -(\sin \theta)^2.
-</math>
-<math>\left( x^{2} -1\right) ^{l}\ =\ (-1)^{l}\left( \sin \theta \right) ^{2l}</math>
-<math>\left( x^{2} -1\right) ^{l-1}\ =\ (-1)^{l-1}\left( \sin \theta \right) ^{2l-2}</math>
-<math>
-\int\limits_{-1}^{1}(x^{2} -1)^{l}  dx \ =\ (-1)^{l+1}\int\limits_{0}^{\pi}\left( \sin \theta \right) ^{2l+1}  d\theta</math>
-<math>
-\int\limits_{-1}^{1}(x^{2} -1)^{l-1}  dx \ =\ (-1)^{l}\int\limits_{0}^{\pi}\left( \sin \theta \right) ^{2l-1}  d\theta</math>
-<math>\int\limits_{0}^{\pi }\sin ^{n} \theta  d\theta  =\frac{\left(
-n-1\right) }{n} \int\limits_{0}^{\pi }\sin ^{n-2} \theta  d\theta
-</math>
-<math>
-\int\limits_{-1}^{1}(x^{2} -1)^{l}  dx \ =\ (-1)^{l+1}\int\limits_{0}^{\pi}\left( \sin \theta \right) ^{2l+1}  d\theta\ =\ (-1)^{l+1} \frac{2l}{2l+1}\int\limits_{0}^{\pi}\left( \sin \theta \right) ^{2l-1}  d\theta\ =\ -\ \frac{2l}{2l+1} \int\limits_{-1}^{1}\left( x^{2} -1\right) ^{l-1}  dx
-</math>