miniaufgabe.js ==== 4. März 2019 bis 8. März 2019 ==== === Mittwoch 6. März 2019 === Berechnen Sie von Hand und ohne Unterlagen: miniAufgabe("#exointpoly","#solintpoly", [["$\\displaystyle \\int_{-\\frac{1}{2}}^{\\frac{1}{2}}\\left(-\\frac{3}{2}x^{2}-\\frac{1}{3}x-\\frac{3}{2}\\right) \\mathrm{d}x$", "$\\left(-\\frac{1}{2}x^{3}-\\frac{1}{6}x^{2}-\\frac{3}{2}x\\right) \\Bigr|_{-\\frac{1}{2}}^{\\frac{1}{2}} = \\left(-\\frac{1}{2}\\cdot \\frac{1}{8}-\\frac{1}{6}\\cdot \\frac{1}{4}-\\frac{3}{2}\\cdot \\frac{1}{2}\\right) - \\left(-\\frac{1}{2}\\cdot \\left(-\\frac{1}{8}\\right)-\\frac{1}{6}\\cdot \\frac{1}{4}-\\frac{3}{2}\\cdot \\left(-\\frac{1}{2}\\right)\\right) = $
$\\left(-\\frac{1}{16}-\\frac{1}{24}-\\frac{3}{4}\\right) - \\left(\\frac{1}{16}-\\frac{1}{24}+\\frac{3}{4}\\right) = \\left(-\\frac{3}{48}-\\frac{2}{48}-\\frac{36}{48}\\right) - \\left(\\frac{3}{48}-\\frac{2}{48}+\\frac{36}{48}\\right) = -\\frac{41}{48} - \\frac{37}{48} = -\\frac{13}{8}$"], ["$\\displaystyle \\int_{-\\frac{2}{3}}^{\\frac{4}{3}}\\left(-\\frac{1}{2}x^{3}+\\frac{2}{3}x+\\frac{1}{2}\\right) \\mathrm{d}x$", "$\\left(-\\frac{1}{8}x^{4}+\\frac{1}{3}x^{2}+\\frac{1}{2}x\\right) \\Bigr|_{-\\frac{2}{3}}^{\\frac{4}{3}} = \\left(-\\frac{1}{8}\\cdot \\frac{256}{81}+\\frac{1}{3}\\cdot \\frac{16}{9}+\\frac{1}{2}\\cdot \\frac{4}{3}\\right) - \\left(-\\frac{1}{8}\\cdot \\frac{16}{81}+\\frac{1}{3}\\cdot \\frac{4}{9}+\\frac{1}{2}\\cdot \\left(-\\frac{2}{3}\\right)\\right) = $
$\\left(-\\frac{32}{81}+\\frac{16}{27}+\\frac{2}{3}\\right) - \\left(-\\frac{2}{81}+\\frac{4}{27}-\\frac{1}{3}\\right) = \\left(-\\frac{32}{81}+\\frac{48}{81}+\\frac{54}{81}\\right) - \\left(-\\frac{2}{81}+\\frac{12}{81}-\\frac{27}{81}\\right) = \\frac{70}{81} - \\left(-\\frac{17}{81}\\right) = \\frac{29}{27}$"], ["$\\displaystyle \\int_{-\\frac{1}{3}}^{\\frac{2}{3}}\\left(\\frac{3}{4}x^{2}+\\frac{2}{3}x-\\frac{3}{4}\\right) \\mathrm{d}x$", "$\\left(\\frac{1}{4}x^{3}+\\frac{1}{3}x^{2}-\\frac{3}{4}x\\right) \\Bigr|_{-\\frac{1}{3}}^{\\frac{2}{3}} = \\left(\\frac{1}{4}\\cdot \\frac{8}{27}+\\frac{1}{3}\\cdot \\frac{4}{9}-\\frac{3}{4}\\cdot \\frac{2}{3}\\right) - \\left(\\frac{1}{4}\\cdot \\left(-\\frac{1}{27}\\right)+\\frac{1}{3}\\cdot \\frac{1}{9}-\\frac{3}{4}\\cdot \\left(-\\frac{1}{3}\\right)\\right) = $
$\\left(\\frac{2}{27}+\\frac{4}{27}-\\frac{1}{2}\\right) - \\left(-\\frac{1}{108}+\\frac{1}{27}+\\frac{1}{4}\\right) = \\left(\\frac{4}{54}+\\frac{8}{54}-\\frac{27}{54}\\right) - \\left(-\\frac{1}{108}+\\frac{4}{108}+\\frac{27}{108}\\right) = -\\frac{5}{18} - \\frac{5}{18} = -\\frac{5}{9}$"], ["$\\displaystyle \\int_{-\\frac{2}{3}}^{\\frac{1}{2}}\\left(-3x^{3}-\\frac{9}{8}x^{2}+\\frac{3}{4}x\\right) \\mathrm{d}x$", "$\\left(-\\frac{3}{4}x^{4}-\\frac{3}{8}x^{3}+\\frac{3}{8}x^{2}\\right) \\Bigr|_{-\\frac{2}{3}}^{\\frac{1}{2}} = \\left(-\\frac{3}{4}\\cdot \\frac{1}{16}-\\frac{3}{8}\\cdot \\frac{1}{8}+\\frac{3}{8}\\cdot \\frac{1}{4}\\right) - \\left(-\\frac{3}{4}\\cdot \\frac{16}{81}-\\frac{3}{8}\\cdot \\left(-\\frac{8}{27}\\right)+\\frac{3}{8}\\cdot \\frac{4}{9}\\right) = $
$\\left(-\\frac{3}{64}-\\frac{3}{64}+\\frac{3}{32}\\right) - \\left(-\\frac{4}{27}+\\frac{1}{9}+\\frac{1}{6}\\right) = \\left(-\\frac{3}{64}-\\frac{3}{64}+\\frac{6}{64}\\right) - \\left(-\\frac{8}{54}+\\frac{6}{54}+\\frac{9}{54}\\right) = 0 - \\frac{7}{54} = -\\frac{7}{54}$"], ["$\\displaystyle \\int_{-\\frac{3}{2}}^{-\\frac{1}{2}}\\left(-\\frac{4}{3}x^{3}-\\frac{1}{2}x^{2}+\\frac{1}{2}\\right) \\mathrm{d}x$", "$\\left(-\\frac{1}{3}x^{4}-\\frac{1}{6}x^{3}+\\frac{1}{2}x\\right) \\Bigr|_{-\\frac{3}{2}}^{-\\frac{1}{2}} = \\left(-\\frac{1}{3}\\cdot \\frac{1}{16}-\\frac{1}{6}\\cdot \\left(-\\frac{1}{8}\\right)+\\frac{1}{2}\\cdot \\left(-\\frac{1}{2}\\right)\\right) - \\left(-\\frac{1}{3}\\cdot \\frac{81}{16}-\\frac{1}{6}\\cdot \\left(-\\frac{27}{8}\\right)+\\frac{1}{2}\\cdot \\left(-\\frac{3}{2}\\right)\\right) = $
$\\left(-\\frac{1}{48}+\\frac{1}{48}-\\frac{1}{4}\\right) - \\left(-\\frac{27}{16}+\\frac{9}{16}-\\frac{3}{4}\\right) = \\left(-\\frac{1}{48}+\\frac{1}{48}-\\frac{12}{48}\\right) - \\left(-\\frac{27}{16}+\\frac{9}{16}-\\frac{12}{16}\\right) = -\\frac{1}{4} - \\left(-\\frac{15}{8}\\right) = \\frac{13}{8}$"], ["$\\displaystyle \\int_{\\frac{1}{2}}^{\\frac{3}{2}}\\left(\\frac{16}{9}x^{3}-\\frac{2}{3}x-\\frac{2}{3}\\right) \\mathrm{d}x$", "$\\left(\\frac{4}{9}x^{4}-\\frac{1}{3}x^{2}-\\frac{2}{3}x\\right) \\Bigr|_{\\frac{1}{2}}^{\\frac{3}{2}} = \\left(\\frac{4}{9}\\cdot \\frac{81}{16}-\\frac{1}{3}\\cdot \\frac{9}{4}-\\frac{2}{3}\\cdot \\frac{3}{2}\\right) - \\left(\\frac{4}{9}\\cdot \\frac{1}{16}-\\frac{1}{3}\\cdot \\frac{1}{4}-\\frac{2}{3}\\cdot \\frac{1}{2}\\right) = $
$\\left(\\frac{9}{4}-\\frac{3}{4}-1\\right) - \\left(\\frac{1}{36}-\\frac{1}{12}-\\frac{1}{3}\\right) = \\left(\\frac{9}{4}-\\frac{3}{4}-1\\right) - \\left(\\frac{1}{36}-\\frac{3}{36}-\\frac{12}{36}\\right) = \\frac{1}{2} - \\left(-\\frac{7}{18}\\right) = \\frac{8}{9}$"], ["$\\displaystyle \\int_{-\\frac{3}{2}}^{\\frac{3}{2}}\\left(-\\frac{4}{3}x^{3}+\\frac{1}{3}x^{2}+\\frac{2}{3}x\\right) \\mathrm{d}x$", "$\\left(-\\frac{1}{3}x^{4}+\\frac{1}{9}x^{3}+\\frac{1}{3}x^{2}\\right) \\Bigr|_{-\\frac{3}{2}}^{\\frac{3}{2}} = \\left(-\\frac{1}{3}\\cdot \\frac{81}{16}+\\frac{1}{9}\\cdot \\frac{27}{8}+\\frac{1}{3}\\cdot \\frac{9}{4}\\right) - \\left(-\\frac{1}{3}\\cdot \\frac{81}{16}+\\frac{1}{9}\\cdot \\left(-\\frac{27}{8}\\right)+\\frac{1}{3}\\cdot \\frac{9}{4}\\right) = $
$\\left(-\\frac{27}{16}+\\frac{3}{8}+\\frac{3}{4}\\right) - \\left(-\\frac{27}{16}-\\frac{3}{8}+\\frac{3}{4}\\right) = \\left(-\\frac{27}{16}+\\frac{6}{16}+\\frac{12}{16}\\right) - \\left(-\\frac{27}{16}-\\frac{6}{16}+\\frac{12}{16}\\right) = -\\frac{9}{16} - \\left(-\\frac{21}{16}\\right) = \\frac{3}{4}$"], ["$\\displaystyle \\int_{-\\frac{3}{2}}^{-\\frac{1}{2}}\\left(-\\frac{2}{3}x^{2}+\\frac{2}{3}x+\\frac{3}{2}\\right) \\mathrm{d}x$", "$\\left(-\\frac{2}{9}x^{3}+\\frac{1}{3}x^{2}+\\frac{3}{2}x\\right) \\Bigr|_{-\\frac{3}{2}}^{-\\frac{1}{2}} = \\left(-\\frac{2}{9}\\cdot \\left(-\\frac{1}{8}\\right)+\\frac{1}{3}\\cdot \\frac{1}{4}+\\frac{3}{2}\\cdot \\left(-\\frac{1}{2}\\right)\\right) - \\left(-\\frac{2}{9}\\cdot \\left(-\\frac{27}{8}\\right)+\\frac{1}{3}\\cdot \\frac{9}{4}+\\frac{3}{2}\\cdot \\left(-\\frac{3}{2}\\right)\\right) = $
$\\left(\\frac{1}{36}+\\frac{1}{12}-\\frac{3}{4}\\right) - \\left(\\frac{3}{4}+\\frac{3}{4}-\\frac{9}{4}\\right) = \\left(\\frac{1}{36}+\\frac{3}{36}-\\frac{27}{36}\\right) - \\left(\\frac{3}{4}+\\frac{3}{4}-\\frac{9}{4}\\right) = -\\frac{23}{36} - \\left(-\\frac{3}{4}\\right) = \\frac{1}{9}$"], ["$\\displaystyle \\int_{\\frac{1}{2}}^{\\frac{3}{2}}\\left(x^{2}-\\frac{1}{3}x+\\frac{1}{9}\\right) \\mathrm{d}x$", "$\\left(\\frac{1}{3}x^{3}-\\frac{1}{6}x^{2}+\\frac{1}{9}x\\right) \\Bigr|_{\\frac{1}{2}}^{\\frac{3}{2}} = \\left(\\frac{1}{3}\\cdot \\frac{27}{8}-\\frac{1}{6}\\cdot \\frac{9}{4}+\\frac{1}{9}\\cdot \\frac{3}{2}\\right) - \\left(\\frac{1}{3}\\cdot \\frac{1}{8}-\\frac{1}{6}\\cdot \\frac{1}{4}+\\frac{1}{9}\\cdot \\frac{1}{2}\\right) = $
$\\left(\\frac{9}{8}-\\frac{3}{8}+\\frac{1}{6}\\right) - \\left(\\frac{1}{24}-\\frac{1}{24}+\\frac{1}{18}\\right) = \\left(\\frac{27}{24}-\\frac{9}{24}+\\frac{4}{24}\\right) - \\left(\\frac{3}{72}-\\frac{3}{72}+\\frac{4}{72}\\right) = \\frac{11}{12} - \\frac{1}{18} = \\frac{31}{36}$"], ["$\\displaystyle \\int_{-\\frac{2}{3}}^{\\frac{2}{3}}\\left(-\\frac{9}{8}x^{2}-\\frac{2}{7}x-\\frac{1}{3}\\right) \\mathrm{d}x$", "$\\left(-\\frac{3}{8}x^{3}-\\frac{1}{7}x^{2}-\\frac{1}{3}x\\right) \\Bigr|_{-\\frac{2}{3}}^{\\frac{2}{3}} = \\left(-\\frac{3}{8}\\cdot \\frac{8}{27}-\\frac{1}{7}\\cdot \\frac{4}{9}-\\frac{1}{3}\\cdot \\frac{2}{3}\\right) - \\left(-\\frac{3}{8}\\cdot \\left(-\\frac{8}{27}\\right)-\\frac{1}{7}\\cdot \\frac{4}{9}-\\frac{1}{3}\\cdot \\left(-\\frac{2}{3}\\right)\\right) = $
$\\left(-\\frac{1}{9}-\\frac{4}{63}-\\frac{2}{9}\\right) - \\left(\\frac{1}{9}-\\frac{4}{63}+\\frac{2}{9}\\right) = \\left(-\\frac{7}{63}-\\frac{4}{63}-\\frac{14}{63}\\right) - \\left(\\frac{7}{63}-\\frac{4}{63}+\\frac{14}{63}\\right) = -\\frac{25}{63} - \\frac{17}{63} = -\\frac{2}{3}$"]], "

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=== Freitag 8. März 2019 === Berechnen Sie von Hand und ohne Unterlagen: miniAufgabe("#exotrigointegrate","#soltrigointegrate", [["$\\displaystyle \\int_{0}^{\\frac{3 \\pi}{2}} \\left(\\frac{1}{2} \\sin(x) -\\frac{1}{6} \\cos(x) \\right)\\mathrm{d}x$", "$\\left( -\\frac{1}{2} \\cos(x) -\\frac{1}{6} \\sin(x) \\right) \\Bigr|_{0}^{\\frac{3 \\pi}{2}} = \\left( -\\frac{1}{2} \\cos\\left(\\frac{3 \\pi}{2}\\right) -\\frac{1}{6} \\sin\\left(\\frac{3 \\pi}{2}\\right) \\right) - \\left( -\\frac{1}{2} \\cos\\left(0\\right) -\\frac{1}{6} \\sin\\left(0\\right) \\right) = \\left( 0 +\\frac{1}{6} \\right) - \\left( -\\frac{1}{2} +0 \\right) = \\frac{1}{6} +\\frac{1}{2} = \\frac{2}{3}$"], ["$\\displaystyle \\int_{\\frac{ \\pi}{2}}^{\\frac{3 \\pi}{2}} \\left(-\\frac{3}{2} \\sin(x) -\\frac{4}{3} \\cos(x) \\right)\\mathrm{d}x$", "$\\left( \\frac{3}{2} \\cos(x) -\\frac{4}{3} \\sin(x) \\right) \\Bigr|_{\\frac{ \\pi}{2}}^{\\frac{3 \\pi}{2}} = \\left( \\frac{3}{2} \\cos\\left(\\frac{3 \\pi}{2}\\right) -\\frac{4}{3} \\sin\\left(\\frac{3 \\pi}{2}\\right) \\right) - \\left( \\frac{3}{2} \\cos\\left(\\frac{ \\pi}{2}\\right) -\\frac{4}{3} \\sin\\left(\\frac{ \\pi}{2}\\right) \\right) = \\left( 0 +\\frac{4}{3} \\right) - \\left( 0 -\\frac{4}{3} \\right) = \\frac{4}{3} +\\frac{4}{3} = \\frac{8}{3}$"], ["$\\displaystyle \\int_{-\\frac{5 \\pi}{2}}^{2\\pi} \\left(\\frac{1}{2} \\sin(x) +\\frac{1}{5} \\cos(x) \\right)\\mathrm{d}x$", "$\\left( -\\frac{1}{2} \\cos(x) +\\frac{1}{5} \\sin(x) \\right) \\Bigr|_{-\\frac{5 \\pi}{2}}^{2\\pi} = \\left( -\\frac{1}{2} \\cos\\left(2\\pi\\right) +\\frac{1}{5} \\sin\\left(2\\pi\\right) \\right) - \\left( -\\frac{1}{2} \\cos\\left(-\\frac{5 \\pi}{2}\\right) +\\frac{1}{5} \\sin\\left(-\\frac{5 \\pi}{2}\\right) \\right) = \\left( -\\frac{1}{2} +0 \\right) - \\left( 0 -\\frac{1}{5} \\right) = -\\frac{1}{2} +\\frac{1}{5} = -\\frac{3}{10}$"], ["$\\displaystyle \\int_{\\frac{3 \\pi}{2}}^{2\\pi} \\left(-\\frac{4}{9} \\sin(x) +\\frac{1}{4} \\cos(x) \\right)\\mathrm{d}x$", "$\\left( \\frac{4}{9} \\cos(x) +\\frac{1}{4} \\sin(x) \\right) \\Bigr|_{\\frac{3 \\pi}{2}}^{2\\pi} = \\left( \\frac{4}{9} \\cos\\left(2\\pi\\right) +\\frac{1}{4} \\sin\\left(2\\pi\\right) \\right) - \\left( \\frac{4}{9} \\cos\\left(\\frac{3 \\pi}{2}\\right) +\\frac{1}{4} \\sin\\left(\\frac{3 \\pi}{2}\\right) \\right) = \\left( \\frac{4}{9} +0 \\right) - \\left( 0 -\\frac{1}{4} \\right) = \\frac{4}{9} +\\frac{1}{4} = \\frac{25}{36}$"], ["$\\displaystyle \\int_{\\pi}^{2\\pi} \\left(-\\frac{1}{3} \\sin(x) -\\frac{4}{9} \\cos(x) \\right)\\mathrm{d}x$", "$\\left( \\frac{1}{3} \\cos(x) -\\frac{4}{9} \\sin(x) \\right) \\Bigr|_{\\pi}^{2\\pi} = \\left( \\frac{1}{3} \\cos\\left(2\\pi\\right) -\\frac{4}{9} \\sin\\left(2\\pi\\right) \\right) - \\left( \\frac{1}{3} \\cos\\left(\\pi\\right) -\\frac{4}{9} \\sin\\left(\\pi\\right) \\right) = \\left( \\frac{1}{3} +0 \\right) - \\left( -\\frac{1}{3} +0 \\right) = \\frac{1}{3} +\\frac{1}{3} = \\frac{2}{3}$"], ["$\\displaystyle \\int_{-\\frac{3 \\pi}{2}}^{\\pi} \\left(\\frac{1}{4} \\sin(x) -\\frac{1}{2} \\cos(x) \\right)\\mathrm{d}x$", "$\\left( -\\frac{1}{4} \\cos(x) -\\frac{1}{2} \\sin(x) \\right) \\Bigr|_{-\\frac{3 \\pi}{2}}^{\\pi} = \\left( -\\frac{1}{4} \\cos\\left(\\pi\\right) -\\frac{1}{2} \\sin\\left(\\pi\\right) \\right) - \\left( -\\frac{1}{4} \\cos\\left(-\\frac{3 \\pi}{2}\\right) -\\frac{1}{2} \\sin\\left(-\\frac{3 \\pi}{2}\\right) \\right) = \\left( \\frac{1}{4} +0 \\right) - \\left( 0 -\\frac{1}{2} \\right) = \\frac{1}{4} +\\frac{1}{2} = \\frac{3}{4}$"], ["$\\displaystyle \\int_{-\\frac{ \\pi}{2}}^{2\\pi} \\left(\\frac{1}{8} \\sin(x) +\\frac{3}{5} \\cos(x) \\right)\\mathrm{d}x$", "$\\left( -\\frac{1}{8} \\cos(x) +\\frac{3}{5} \\sin(x) \\right) \\Bigr|_{-\\frac{ \\pi}{2}}^{2\\pi} = \\left( -\\frac{1}{8} \\cos\\left(2\\pi\\right) +\\frac{3}{5} \\sin\\left(2\\pi\\right) \\right) - \\left( -\\frac{1}{8} \\cos\\left(-\\frac{ \\pi}{2}\\right) +\\frac{3}{5} \\sin\\left(-\\frac{ \\pi}{2}\\right) \\right) = \\left( -\\frac{1}{8} +0 \\right) - \\left( 0 -\\frac{3}{5} \\right) = -\\frac{1}{8} +\\frac{3}{5} = \\frac{19}{40}$"], ["$\\displaystyle \\int_{\\pi}^{2\\pi} \\left(\\frac{2}{5} \\sin(x) +\\frac{2}{7} \\cos(x) \\right)\\mathrm{d}x$", "$\\left( -\\frac{2}{5} \\cos(x) +\\frac{2}{7} \\sin(x) \\right) \\Bigr|_{\\pi}^{2\\pi} = \\left( -\\frac{2}{5} \\cos\\left(2\\pi\\right) +\\frac{2}{7} \\sin\\left(2\\pi\\right) \\right) - \\left( -\\frac{2}{5} \\cos\\left(\\pi\\right) +\\frac{2}{7} \\sin\\left(\\pi\\right) \\right) = \\left( -\\frac{2}{5} +0 \\right) - \\left( \\frac{2}{5} +0 \\right) = -\\frac{2}{5} -\\frac{2}{5} = -\\frac{4}{5}$"], ["$\\displaystyle \\int_{-2\\pi}^{\\pi} \\left(-\\frac{4}{3} \\sin(x) -\\frac{1}{7} \\cos(x) \\right)\\mathrm{d}x$", "$\\left( \\frac{4}{3} \\cos(x) -\\frac{1}{7} \\sin(x) \\right) \\Bigr|_{-2\\pi}^{\\pi} = \\left( \\frac{4}{3} \\cos\\left(\\pi\\right) -\\frac{1}{7} \\sin\\left(\\pi\\right) \\right) - \\left( \\frac{4}{3} \\cos\\left(-2\\pi\\right) -\\frac{1}{7} \\sin\\left(-2\\pi\\right) \\right) = \\left( -\\frac{4}{3} +0 \\right) - \\left( \\frac{4}{3} +0 \\right) = -\\frac{4}{3} -\\frac{4}{3} = -\\frac{8}{3}$"], ["$\\displaystyle \\int_{-\\pi}^{-\\frac{ \\pi}{2}} \\left(-\\frac{4}{5} \\sin(x) +\\frac{2}{7} \\cos(x) \\right)\\mathrm{d}x$", "$\\left( \\frac{4}{5} \\cos(x) +\\frac{2}{7} \\sin(x) \\right) \\Bigr|_{-\\pi}^{-\\frac{ \\pi}{2}} = \\left( \\frac{4}{5} \\cos\\left(-\\frac{ \\pi}{2}\\right) +\\frac{2}{7} \\sin\\left(-\\frac{ \\pi}{2}\\right) \\right) - \\left( \\frac{4}{5} \\cos\\left(-\\pi\\right) +\\frac{2}{7} \\sin\\left(-\\pi\\right) \\right) = \\left( 0 -\\frac{2}{7} \\right) - \\left( -\\frac{4}{5} +0 \\right) = -\\frac{2}{7} +\\frac{4}{5} = \\frac{18}{35}$"]], "

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