Integral progress and stuff

ma5/uppg/main.aux

@@ -24,4 +24,4 @@
 \@writefile{toc}{\contentsline {section}{\numberline {2}Uppgiftlösningar}{3}{section.2}\protected@file@percent }
 \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}1}{3}{subsection.2.1}\protected@file@percent }
 \newlabel{psi_gen}{{4}{3}{1}{equation.2.4}{}}
-\gdef \@abspage@last{4}
+\gdef \@abspage@last{5}

ma5/uppg/main.fdb_latexmk

@@ -1,5 +1,5 @@
 # Fdb version 3
-["pdflatex"] 1648207301 "main.tex" "main.pdf" "main" 1648207302
+["pdflatex"] 1648211808 "main.tex" "main.pdf" "main" 1648211809
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@@ -9,11 +9,13 @@
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   "/usr/share/texmf-dist/fonts/tfm/public/cm/cmbx12.tfm" 1640600964 1324 c910af8c371558dc20f2d7822f66fe64 ""
+  "/usr/share/texmf-dist/fonts/tfm/public/cm/cmbxti10.tfm" 1640600964 1532 9162035f4e7176612125649e348e2195 ""
   "/usr/share/texmf-dist/fonts/tfm/public/cm/cmr12.tfm" 1640600964 1288 655e228510b4c2a1abe905c368440826 ""
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+  "/usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmbxti10.pfb" 1640600964 36554 b67dc2cfa451409e100b3fcf5f506509 ""
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@@ -79,9 +81,9 @@
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-  "main.aux" 1648207302 1487 a4ec684567e5177a4979c66f29306855 "pdflatex"
+  "main.aux" 1648211809 1487 653c38bcf6a6df8e18201b53f41b3454 "pdflatex"
-  "main.out" 1648207302 561 7f4a6e130b4a0841f3a3fe085e8f354a "pdflatex"
+  "main.out" 1648211809 561 7f4a6e130b4a0841f3a3fe085e8f354a "pdflatex"
-  "main.tex" 1648207301 8477 1a6c71bcad5833f57176814b77d70aaf ""
+  "main.tex" 1648211807 9607 ed2c06c8caf7132ab4ead7927c510590 ""
   (generated)
   "main.aux"
   "main.log"

ma5/uppg/main.fls

@@ -521,10 +521,12 @@ INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmbx12.tfm
 INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmr12.tfm
 INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmbx12.tfm
 INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmti10.tfm
+INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmbxti10.tfm
 INPUT main.aux
 INPUT ./main.out
 INPUT ./main.out
 INPUT /usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb
+INPUT /usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmbxti10.pfb
 INPUT /usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb
 INPUT /usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb
 INPUT /usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb

ma5/uppg/main.log

@@ -1,4 +1,4 @@
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ma5/uppg/main.tex

@@ -210,11 +210,44 @@ $$
 \implies \int_0^L \shrodprob dx\ = D^2 \int_0^L \sin^2\left( \frac{\eta\pi}{L} x \right) dx\ = 1.0
 $$
 
-Vi behöver nu beräkna integralen och få fram dess uttryck. Vi använder oss därmed av \emph{u-substitution} och \emph{trigonometriska ettan}:
+Vi behöver nu beräkna integralen och få fram dess uttryck. Vi använder oss därmed av \emph{u-substitution} och \emph{trigonometriska ettan}. \emph{\textbf{Låt $u = \frac{\eta\pi}{L}x$}}:
-$$ %TODO: remove me
+$$
-2 \sin^2 x = 1 - \cos x
+\int_0^L \sin^2\left( \frac{\eta\pi}{L} x \right) dx\ = \int_0^L \sin^2(u) dx\
+$$
+$$
+\implies \frac{du}{dx} = \frac{\eta\pi}{L} \implies dx\ = \frac{L}{\eta\pi} du\
+$$
+$$
+\implies \int_0^L \sin^2(u) dx\ = \frac{L}{\eta\pi} \int_0^L sin^2(u) du\
+$$
+
+Väljer att \emph{\textbf{skriva om och förenkla $sin^2\left(u\right)$}} och enlight den \emph{trigonometriska ettan} får vi:
+$$
+\cos(2u) = 1 - 2\sin^2 u, \quad \text{(dubbla vinkeln för cosinus)}, \quad +2\sin^2 u
+$$
+$$
+\cos 2u + 2\sin^2 u = 1, \quad -\cos 2u
+$$
+$$
+2\sin^2 u = 1 - \cos 2u, \quad /2
+$$
+$$
+\implies \sin^2 u = \frac{1}{2} - \frac{1}{2}\cos 2u
 $$
 
+Vi kan nu stoppa in vår förenklade version av $sin^2(u)$ med något vi faktiskt kan integrera:
+$$
+\implies \int_0^L \shrodprob dx\ 
+= D^2 \int_0^L \sin^2\left( \frac{\eta\pi}{L} x \right) dx\ 
+= D^2 \frac{L}{\eta\pi} \int_0^L \left\{ \frac{1}{2} - \frac{1}{2}\cos(2u) \right\} du\ 
+$$
+$$
+= D^2 \frac{L}{2\eta\pi} \int_0^L \left\{ 1 - \cos(2u) \right\} du\ 
+= D^2 \frac{L}{2\eta\pi} \left( \int_0^L 1 du\ - \int_0^L \cos(2u) du\ \right)
+$$
+$$
+= D^2 \frac{L}{2\eta\pi} \left[u - \int \cos(2u) du\ \right]_0^L
+$$