Unendliche Integrale Partielle Integration
Skript - Ttp-schreiber.de
Skript - Ttp-schreiber.de
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∫ sin( dx = F1<br />
( x)<br />
+ C<br />
2x<br />
1 2x<br />
f ′(<br />
x)<br />
= e ⇒ f ( x)<br />
= ⋅e<br />
2<br />
1. Schritt: 1(<br />
x)<br />
=<br />
2x<br />
Beispiel zu Fall 2: ( x)<br />
⋅e<br />
)<br />
g(<br />
x)<br />
= sin( x)<br />
⇒<br />
2<br />
( ⋅ x)<br />
)<br />
g′<br />
( x)<br />
= cos( x)<br />
1 2<br />
F 1 ⎛<br />
e 2 x ⋅sin(<br />
x ) − e x ⎞<br />
∫ ⎜ ⋅ cos( x)<br />
⎟dx<br />
2<br />
⎝ 2 ⎠<br />
∫ e x cos( dx = F2<br />
( x)<br />
+ C<br />
2x<br />
1 2x<br />
f ′(<br />
x)<br />
= e ⇒ f ( x)<br />
= ⋅e<br />
2<br />
2. Schritt: F<br />
2<br />
( x)<br />
=<br />
1 e 2 x ⎛ 1<br />
⋅cos(<br />
x ) − e 2 x ⎞<br />
2<br />
∫ ⎜ ⋅ ( −sin(<br />
x))<br />
⎟dx<br />
⎝ 2<br />
⎠<br />
g(<br />
x)<br />
= cos( x)<br />
⇒ g′<br />
( x)<br />
= −sin(<br />
x)<br />
SS 2013<br />
⇒<br />
⇔<br />
∫<br />
∫<br />
2x<br />
1 2x<br />
1 ⎛ 1 2x<br />
1 2x<br />
( sin( x)<br />
⋅e<br />
) dx = e ⋅sin(<br />
x)<br />
− ⎜ e ⋅cos(<br />
x)<br />
+ ( e ⋅sin(<br />
x)<br />
)<br />
2<br />
2x<br />
1 2x<br />
⎛ 1 ⎞ 1 2x<br />
( sin( x)<br />
⋅e<br />
) dx = e ⋅⎜sin(<br />
x)<br />
− cos( x)<br />
⎟ − ( e ⋅sin(<br />
x)<br />
)<br />
2<br />
⎝<br />
2 ⎝ 2<br />
2<br />
⎠<br />
g(x)=alternierend; f(x)=alternierend<br />
2x<br />
1 2x<br />
⎛ 1 ⎞<br />
2x<br />
2 x ⎛ 1 ⎞<br />
( sin( x)<br />
⋅e<br />
) dx = e ⋅⎜sin(<br />
x)<br />
− cos( x)<br />
⎟ ⇔ ( sin( x)<br />
⋅e<br />
) dx = e ⋅⎜sin(<br />
x)<br />
− cos( x ⎟<br />
⎠<br />
5 2<br />
∫ ∫<br />
)<br />
4<br />
2 ⎝ 2 ⎠<br />
5 ⎝ 2<br />
4<br />
∫<br />
2<br />
∫<br />
⎞<br />
dx⎟<br />
⎠<br />
Torsten Schreiber 185<br />
dx
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