Jan Math 2008-12-05

allgemein

homogene DGL
$ay''+by'+cy=0$ (oder auch höhere Ableitungen von y)
inhomogene DGL
$ay''+by'+cy=f(x)$
Gelöst witd zuerst die homogene DGL - die Lösung der inhomogenen ist eine (irgend eine) Löung der inhomogenen plus die allgemeine Löung der homogenen
$y=y_{sp}+y_{h}$

- 4.1 d)

$y''+6x=0$
- homogene -
$y''=0$
D.h. y zweimal differenziert ist 0, da kann y maximal x sein (Polynom ersten Grades). Homogene Lösung (allgemein)
$y_{h}=ax+b$
- spezielle Lösung -
$y_{sp}''=-6x$
Einfach zweimal integrieren:
$y_{sp}'=-3x^{2}$
(kein +C, da man ja nur eine spezielle Lösung sucht!)
$y_{sp}=-x^{3}$
- Gesamtlösung -
$y=y_{sp}+y_{h}$
$y=-x^{3}+ax+b$
(a,b beliebig)

- 4.1 e)

$y''+6x-3=0$
- homogene -
$y_{h}''=0$
$y_{h}=ax+b$
- spezielle -
$y_{sp}''=-6x+3$
$y_{sp}'=-3x^{2}+3x$
$y_{sp}=-x^{3}+{3 \over 2}x^{2}$
- Gesamtlösung -
$y=y_{sp}+y_{h}$
$y=-x^{3}+{3 \over 2}x^{2}+ax+b$

- 4.1 f)

$y''+12x^{2}-4x=1$
- homogene (wie schon zwei Mal) -
$y_{h}=ax+b$
- spezeille -
$y_{sp}''=-12x^{2}+4x+1$
$y_{sp}'=-4x^{3}+2x^{2}+x$
$y_{sp}=-x^{4}+{2 \over 3}x^{3}+{1 \over 2}x^{2}$
$y_{sp2}=-x^{4}+{2 \over 3}x^{3}$
- Gesmatlösung -
$y=-x^{4}+{2 \over 3}x^{3}+{1 \over 2}x^{2}+ax+b$

- 4.2 c)

$y''-x+1=0$
$y(1)=0$
$y'(1)=0$
- allgemein -
$y''=x-1$
$y_{h}''=0$
$y_{h}=ax+b$
$y_{sp}''=x-1$
$y_{sp}'={x^{2} \over 2}-x$
$y_{sp}={x^{3} \over 6}-{x^{2} \over 2}$
$y={x^{3} \over 6}-{x^{2} \over 2}+ax+b$ [1]
$y'={x^{2} \over 2}-x+a$ [2]
Jetzt in [2] laut Anfangsbedingung einsetzen:
$0={1 \over 2}-1+a$
$a={1 \over 2}$
Jetzt in [1] laut Anfangsbedingung einsetzen:
$0={1 \over 6}-{1 \over 2}+a+b$
$0={1 \over 6}-{1 \over 2}+{1 \over 2}+b$
$b=-{1 \over 6}$
$y={x^{3} \over 6}-{x^{2} \over 2}+{1 \over 2}x-{1 \over 6}$

- 4.2 d)

$y''+6x^{2}=1-x$
$y(0)=2$
$y'(0)=5$
- allgemein -
$y''=-6x^{2}-x+1$
$y_{h}=ax+b$
$y_{sp}''=-6x^{2}-x+1$
$y_{sp}'=-2x^{3}-{1 \over 2}x^{2}+x$
$y_{sp}=-{1 \over 2}x^{4}-{1 \over 6}x^{3}+{1 \over 2}x^{2}$
$y=-{1 \over 2}x^{4}-{1 \over 6}x^{3}+{1 \over 2}x^{2}+ax+b$ [1]
$y'=-2x^{3}-{1 \over 2}x^{2}+x+a$ [2]
Einsetzen in [2]
$5=0+0+0+a$
$a=5$
Einsetzen in [1]
$2=0+0+0+0+b$
$b=2$
$y=-{1 \over 2}x^{4}-{1 \over 6}x^{3}+{1 \over 2}x^{2}+5x+2$

- 4.3 a)

$y''=2x-1$
$y(0)=1$
$y(6)=1$
$y_{h}=ax+b$
$y_{sp}''=2x-1$
$y_{sp}'=x^{2}-x$
$y_{sp}={1 \over 3}x^{3}-{1 \over 2}x^{2}$
$y={1 \over 3}x^{3}-{1 \over 2}x^{2}+ax+b$
Einsetzen y(0)=1
$1=0-0+0+b$
$b=1$
Einsetzen y6)=1
$1={1 \over 3}6^{3}-{1 \over 2}6^{2}+a6+b$
$1=72-18+6a+1$
$-54=6a$
$a=9$
$y={1 \over 3}x^{3}-{1 \over 2}x^{2}+9x+1$

- 4.3 b)

$y''-x+3=0$
$y(3)=1$
$y(9)=10$
$y_{h}=ax+b$
$y_{sp}''=x-3$
$y_{sp}'={1 \over 2}x^{2}-3x$
$y_{sp}={1 \over 6}x^{3}-{3 \over 2}x^{2}$
$y={1 \over 6}x^{3}-{3 \over 2}x^{2}+ax+b$
Einsetzen => lineares Gleichungssystem:
$1={1 \over 6}3^{3}-{3 \over 2}3^{2}+a3+b$
$1={27 \over 6}-{27 \over 2}+3a+b$
$3a+b={6 \over 6}-{27 \over 6}+{81 \over 6}={60 \over 6}=10$ [1]
$1={1 \over 6}9^{3}-{3 \over 2}9^{2}+a9+b$
$1={729 \over 6}-{243 \over 2}+9a+b$
$9a+b={6 \over 6}-{729 \over 6}+{729 \over 6}=1$ [2]
Subtrahiere [1] von [2]
$6a=1-10=-9$
$a=-{9 \over 6}=-{3 \over 2}$
Einsetzen in [1]
$3(-{3 \over 2})+b=10$
$b=10+{9 \over 2}={29 \over 2}$
$y={1 \over 6}x^{3}-{3 \over 2}x^{2}-{3 \over 2}x+{29 \over 2}$

- 4.4

Vorläufig aufgeschoben... $M_{b}$ ? Biegung ?

-4.7 g)

$y'-y^{2}cosx=0$
$y'=y^{2}cosx$
${dy \over dx}=y^{2}cosx$ (* dx / $y^{2}$)
${1 \over {y^{2}}}dy=cosxdx$ (integrieren)
${-1 \over y}=sinx+c$
$y={-1 \over {sinx+c}}$
Probe:
$y'={cosx \over {(sinx+c)^{2}}}$
$y^{2}={1 \over {(sinx+c)^{2}}}$
$y'=y^{2}cosx$ passt

- 4.7 h)

$y'cosx+ysinx=0$
$y'cosx=-ysinx$
${dy \over dx}cosx=-ysinx$ (* dx / cosx / y)
${1 \over y}dy={-sinx \over cosx}dx$ (integrieren)

$lny=-ln({2 \over {1+cosx}})+ln(-{{2cosx} \over {1+cosx}})+c_{1}=ln(cosx)+c_{1}$ (e^ $c_{2}=e^{c_{1}}$)
$y=c_{2}cosx$
Probe:
$y'=c_{2}-sinx$
$y'cosx=c_{2}-sinxcosx$
$ysinx=c_{2}sinxcosx$
$y'cosx+ysinx=c_{3}(-sinxcosx+sinxcosx)=0$

-4.7 i)

$y'sinx+y=0$
$y'sinx=-y$
${dy \over dx}sinx=-y$ (* dx / y / sinx)
${1 \over y}dy={-1 \over sinx}dx$ (integrieren)
$lny=-ln({sinx \over {1+cosx}})+c_{1}$ (e^ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2=e^{c_1}} )
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=c_2 {{1 + cosx} \over sinx}}
Probe:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=c_2 {{-sinx sinx - (1+cosx) cosx} \over {sin^2 x}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=c_2{{-sin^2 x -cosx -cos^2 x} \over {sin^2 x}}} (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sin^2x+cos^2x = 1} )
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=c_2{{-1 -cosx} \over {sin^2 x}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' sinx = -y} passt

-4.8 g)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x y' + x y = y}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(1)=1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x y' = y (1-x)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x {dy \over dx} = y (1-x)} (* dx / x / y)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1 \over y} dy = {{1-x} \over x} dx} (integrieren)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle lny = lnx -x +c_1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=c_2 {x \over {e^x}}}
Probe:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=c_2{{1 e^x - x e^x \over (e^x)^2}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=c_2{{e^x (1-x)} \over {e^x e^x}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=c_2{(1-x) \over e^x}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x y'= c_2{{x(1-x)} \over e^x}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x y = c_2 {x^2 \over e^x}}
$xy'+xy=c_{2}{{x(1-x)+x^{2}} \over e^{x}}$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x y' + x y = c_2 {{x -x^2 + x^2} \over e^x}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x y' + x y = c_2 {x \over e^x}} passt
Ensetzen y(1)=1
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1=c_2 {1 \over {e^1}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2=e}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=e{x \over {e^x}}}

- 4.8 h)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' \sqrt{x} + 2y=1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(0)=1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {dy \over dx} \sqrt{x} = 1 - 2y} (* dx / sqrt(x) / (1-2y))
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle { 1 \over {1-2y}} dy = { 1 \over \sqrt{x}} dx = x^{-{1 \over 2}} dx} (integrieren)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -{{ln(2y - 1)} \over 2} = 2 \sqrt{x} + c_1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -ln(2y - 1) = 4 \sqrt{x} + c_2} (e^ )
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1 \over {2y -1}} = c_3 e^{4 x^{1/2}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2y -1 = {c_4 \over e^{4 x^{1/2}}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2y = {c_4 \over e^{4 x^{1/2}}} + 1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = {c_4 \over {2 e^{4 x^{1/2}}}} + {1 \over 2}}
Probe:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'={ {-c_4 \cdot 2 \cdot 4 \cdot {1 \over 2} x^{-1 \over 2} {e^{4 x^{1/2}}} } \over {({2 e^{4 x^{1/2}})}^2} }}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'={ {-c_4 \cdot 4 \cdot {1 \over 2} x^{-1 \over 2} } \over {2 e^{4 x^{1/2}}} }}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'={ {-c_4 \cdot x^{-1 \over 2} } \over {e^{4 x^{1/2}}} }}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' \sqrt{x}={ {-c_4 \cdot x^{-1 \over 2} \cdot x^{1 \over 2} } \over {e^{4 x^{1 \over 2}}} }}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ={ -c_4 \over {e^{4 x^{1 \over 2}}} }}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' \sqrt{x} + 2y={ {-c_4 \over {e^{4 x^{1 \over 2}}}} + {c_4 \over e^{4 x^{1/2}}} + 1}=1} passt

- 6.8 h)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int {1 \over {sin^2 x}} \, dx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ({u \over v})'={{u'v - u v'} \over {v^2}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=sinx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u'v - u v'= 1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v^2=sin^2 x}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v'=cosx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u' sinx - u cosx = 1} (das klappt mit u'=sinx und u=-cosx)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=-cosx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int {1 \over {sin^2 x}} \, dx = {-cosx \over sinx} +c = -{cosx \over sinx} +c}

- 6.8 i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int {1 \over {1+x^2}}\, dx}
?

- 6.9 e)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)={1 \over x}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(e/4)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(e)=4}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)=lnx + c}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle lne+c=4}
$1+c=4$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c=3}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)=lnx+3}

- 6.9 f)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=2^x}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(1/3)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(1)=3}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=e^{x ln(2)}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)={1 \over ln2}e^{x ln(2)}+c}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3={1 \over ln2}e^{1 ln(2)}+c}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3={2 \over ln2}+c}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c=3-{2 \over ln2}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)={1 \over ln2}e^{x ln(2)}+3-{2 \over ln2}}

- 6.22 d)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int { 1 \over \sqrt {1-x}} \, dx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=1-x}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {du \over dx}=-1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx=-dx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int { 1 \over \sqrt {1-x}} \, dx= -\int {1 \over \sqrt {u}}\, du=-\int {u^{-{1 \over 2}}}\, du}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-{1 \over 2} u^{1\over2}+c}
$=-{1 \over 2}(1-x)^{1 \over 2}+c$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-{1 \over 2} \sqrt{1-x}+c}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ={- \sqrt{1-x} \over 2}+c}

- 6.23 d)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int { 2a \over {a+2x}} \, dx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=a+2x}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {du \over dx}=2}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx={du \over 2}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int { 2a \over {a+2x}} \, dx={1 \over 2} \int {2a \over u} \, du}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ={2a \over 2} lnu+c=a \cdot lnu+c}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =a \cdot ln(a+2x)+c}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =ln((a+2x)^a)+c}

- 6.24 d)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int { 2 \over \sqrt[3] {4x-1}} \, dx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=4x-1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {du \over dx} = 4}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx = {du \over 4}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int { 2 \over \sqrt[3] {4x-1}} \, dx = {1 \over 4} \int {2 \over \sqrt[3]{u}} \, du={2 \over 4} \int {u^{-{1 \over 3}}} \, du={1 \over 2} \int {u^{-{1 \over 3}}} \, du}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ={1 \over 2} \cdot {2 \over 3} u^{2 \over 3}+c}
$={2 \over 6}(4x-1)^{2 \over 3}+c$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ={1 \over 3} \sqrt[3]{(4x-1)^2}+c}

- 6.25 d)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int (e^{3x} - e^{-3x}) \, dx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int {e^{3x}} \,dx - \int {e^{-3x}} \, dx }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ={1 \over 3} e^{3x} - {-1 \over 3} e^{-3x}+c}
$={{e^{3x}+e^{-3x}} \over 3}+c$