More integration

ma5/uppg/main.fdb_latexmk

@@ -1,5 +1,5 @@
 # Fdb version 3
-["pdflatex"] 1648211808 "main.tex" "main.pdf" "main" 1648211809
+["pdflatex"] 1648213365 "main.tex" "main.pdf" "main" 1648213366
   "/usr/share/texmf-dist/fonts/map/fontname/texfonts.map" 1640600964 3524 cb3e574dea2d1052e39280babc910dc8 ""
   "/usr/share/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1640600964 1004 54797486969f23fa377b128694d548df ""
   "/usr/share/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1640600964 916 f87d7c45f9c908e672703b83b72241a3 ""
@@ -13,6 +13,7 @@
   "/usr/share/texmf-dist/fonts/tfm/public/cm/cmr12.tfm" 1640600964 1288 655e228510b4c2a1abe905c368440826 ""
   "/usr/share/texmf-dist/fonts/tfm/public/cm/cmr17.tfm" 1640600964 1292 296a67155bdbfc32aa9c636f21e91433 ""
   "/usr/share/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1640600964 1480 aa8e34af0eb6a2941b776984cf1dfdc4 ""
+  "/usr/share/texmf-dist/fonts/tfm/public/cm/cmti7.tfm" 1640600964 1492 86331993fe614793f5e7e755835c31c5 ""
   "/usr/share/texmf-dist/fonts/tfm/public/cm/cmtt10.tfm" 1640600964 768 1321e9409b4137d6fb428ac9dc956269 ""
   "/usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb" 1640600964 32080 340ef9bf63678554ee606688e7b5339d ""
   "/usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmbxti10.pfb" 1640600964 36554 b67dc2cfa451409e100b3fcf5f506509 ""
@@ -81,9 +82,9 @@
   "/usr/share/texmf-dist/web2c/texmf.cnf" 1640600964 39911 2da6c67557ec033436fe5418a70a8a61 ""
   "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1647853768 4243317 6cd4c1d3b72728bfa7db8ef2e14c0430 ""
   "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1647853732 2920113 d8a499937a1239664546ceed552c7509 ""
-  "main.aux" 1648211809 1487 653c38bcf6a6df8e18201b53f41b3454 "pdflatex"
-  "main.out" 1648211809 561 7f4a6e130b4a0841f3a3fe085e8f354a "pdflatex"
-  "main.tex" 1648211807 9607 ed2c06c8caf7132ab4ead7927c510590 ""
+  "main.aux" 1648213365 1487 653c38bcf6a6df8e18201b53f41b3454 "pdflatex"
+  "main.out" 1648213365 561 7f4a6e130b4a0841f3a3fe085e8f354a "pdflatex"
+  "main.tex" 1648213364 10029 189ca3e9f10596bb1446685a27f92be0 ""
   (generated)
   "main.aux"
   "main.log"

ma5/uppg/main.fls

@@ -522,6 +522,10 @@ 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 /usr/share/texmf-dist/fonts/tfm/public/cm/cmti7.tfm
+INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmbxti10.tfm
+INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmti7.tfm
+INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmbxti10.tfm
 INPUT main.aux
 INPUT ./main.out
 INPUT ./main.out

ma5/uppg/main.log

@@ -1,4 +1,4 @@
-This is pdfTeX, Version 3.141592653-2.6-1.40.23 (TeX Live 2021/Arch Linux) (preloaded format=pdflatex 2022.3.21)  25 MAR 2022 13:36
+This is pdfTeX, Version 3.141592653-2.6-1.40.23 (TeX Live 2021/Arch Linux) (preloaded format=pdflatex 2022.3.21)  25 MAR 2022 14:02
 entering extended mode
  restricted \write18 enabled.
  file:line:error style messages enabled.
@@ -383,15 +383,15 @@ Package rerunfilecheck Info: File `main.out' has not changed.
 (rerunfilecheck)             Checksum: 7F4A6E130B4A0841F3A3FE085E8F354A;561.
  ) 
 Here is how much of TeX's memory you used:
- 9098 strings out of 478353
- 140920 string characters out of 5854635
+ 9103 strings out of 478353
+ 140983 string characters out of 5854635
  463010 words of memory out of 5000000
- 27118 multiletter control sequences out of 15000+600000
- 407818 words of font info for 44 fonts, out of 8000000 for 9000
+ 27122 multiletter control sequences out of 15000+600000
+ 409234 words of font info for 48 fonts, out of 8000000 for 9000
  1141 hyphenation exceptions out of 8191
  60i,7n,63p,809b,330s stack positions out of 5000i,500n,10000p,200000b,80000s
 </usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmbxti10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm5.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm7.pfb>
-Output written on main.pdf (5 pages, 188516 bytes).
+Output written on main.pdf (5 pages, 189172 bytes).
 PDF statistics:
  147 PDF objects out of 1000 (max. 8388607)
  104 compressed objects within 2 object streams

ma5/uppg/main.tex

@@ -249,6 +249,24 @@ $$
 = D^2 \frac{L}{2\eta\pi} \left[u - \int \cos(2u) du\ \right]_0^L
 $$
 
+$$
+\emph{\textbf{Låt $\mu = 2u$}}
+$$
+$$
+\implies \int \cos(2u) du\ = \int \cos(\mu) du\
+$$
+$$
+\implies \frac{d\mu}{du} = 2 \implies du\ = \frac{1}{2} d\mu\
+$$
+$$
+\int \cos(2u) du\ = \frac{1}{2} \int \cos(\mu) d\mu\ = -\frac{1}{2} \sin(\mu) + C
+$$
+
+$$
+\implies \int_0^L \shrodprob dx\ 
+= D^2 \frac{L}{2\eta\pi} \left[u - \int \cos(2u) du\ \right]_0^L
+= D^2 \frac{L}{2\eta\pi} \left[ u + \frac{1}{2}\sin \mu \right]_0^L
+$$
 
 
 \end{document}