Name: using substitution to evaluate definite integrals
Uploaded: 2026-01-12T12:24:59.000Z
Duration: 1 min 24 s
Channel: JoVE

Betrachten wir ein Integral, dessen Argument als Kettenregelableitung einer zusammengesetzten Funktion F(g(x)) geschrieben werden kann. Um das Integral zu lösen, besteht der Prozess darin, die Kettenregel-Differenzierung umzukehren.Hier wird eine neue Variable, u, als g(x) definiert. Dann differenziere u bezüglich x. Dies kann in Bezug auf du umgeordnet werden.Um die Grenzwerte zu ändern: Wenn x=a, u gleich g(a) der neue untere Grenzwert wird, und wenn x=b gleich g(b) der neue obere Grenzwert wird.Das ursprüngliche Integral wird dann umgeschrieben, indem u und du eingesetzt und die Grenzwerte a und b durch g(a) bzw. g(b) ersetzt werden. Die Integration des neuen Integranden bezüglich U und Anwendung der sich ändernden Grenzwerte ergibt den endgültigen numerischen Ausdruck.Ein Beispiel für Substitution findet sich in der Elektrotechnik, wo sie verwendet wird, um die Gesamtladung zu bestimmen, die über einen bestimmten Zeitraum durch einen Stromkreis geleitet wird. Hier wird die Gesamtladung ermittelt, indem das definite Integral bezüglich der Zeit berechnet wird. Da der Strom durch eine komplexe Funktion gegeben wird, erleichtert die Substitution das Integral leichter zu lösen.

substitutionsregel angewandt auf bestimmte integrale

Bei der Berechnung eines bestimmten Integrals, dessen Integrand die Struktur einer zusammengesetzten Funktion hat, bietet die Substitutionsmethode eine effiziente M&#xF6;glichkeit zur Vereinfachung der Berechnung. Diese Methode beruht auf der Umkehrung der Kettenregel der Differentiation und erm&#xF6;glicht es, einen komplizierten Ausdruck in eine einfachere Form umzuformen. Enth&#xE4;lt der Integrand eine innere Funktion und deren Ableitung, reduziert die Substitution die Komplexit&#xE4;t des Problems auf nat&#xFC;rliche Weise.Die zentrale Idee der Substitution f&#xFC;r bestimmte Integrale wird durch die folgende Beziehung zusammengefasst:\begin{equation*}\int_a^b f(g(x))\,g&#x27;(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du\\end{equation*}Der Prozess beginnt mit der Identifizierung der inneren Funktion und der Einf&#xFC;hrung einer neuen Variablen zu ihrer Darstellung. Die Ableitung dieser neuen Variablen liefert ein entsprechendes Differential, das das urspr&#xFC;ngliche ersetzt. Gleichzeitig werden die Integrationsgrenzen ge&#xE4;ndert, indem die innere Funktion an den urspr&#xFC;nglichen Endpunkten ausgewertet wird. So wird sichergestellt, dass das gesamte Integral konsistent in Bezug auf die neue Variable ausgedr&#xFC;ckt wird, ohne dass nach der Integration eine R&#xFC;ckkehr zur urspr&#xFC;nglichen Variablen erforderlich ist.Nach Abschluss der Substitution l&#xE4;sst sich das Integral einfacher auswerten, da der Integrand nun nur von einer Variablen abh&#xE4;ngt. Die direkte Anwendung der angepassten Integrationsgrenzen liefert den endg&#xFC;ltigen numerischen Wert. Dieses Verfahren erleichtert Berechnungen, die mit herk&#xF6;mmlichen Integrationsmethoden sonst schwierig oder unpraktisch w&#xE4;ren.Die Substitution spielt in zahlreichen Anwendungsbereichen eine entscheidende Rolle, einschlie&#xDF;lich der Elektrotechnik. Beispielsweise erfordert die Bestimmung der Gesamtladung, die in einem Stromkreis &#xFC;ber ein bestimmtes Zeitintervall flie&#xDF;t, die Integration des Stroms &#xFC;ber dieses Zeitintervall. Wird der Strom durch einen komplexen Ausdruck beschrieben, erm&#xF6;glicht die Substitution eine effiziente Berechnung des Integrals und stellt damit ein praktisches und wirkungsvolles Werkzeug zur Analyse realer Systeme dar.

Substitutionsregel angewandt auf bestimmte Integrale

Bei der Berechnung eines bestimmten Integrals, dessen Integrand die Struktur einer zusammengesetzten Funktion hat, bietet die Substitutionsmethode eine ...

substitution-rule-applied-to-definite-integrals

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Substitution Rule Applied to Definite Integrals

using substitution to evaluate definite integrals

Substitution makes definite integrals easier to evaluate when the integrand is a composite function. It works by reversing the chain rule from differentiation. When the expression includes an inner function and its derivative, substitution can simplify the setup quickly.

The method starts by choosing the inner function and naming it with a new variable. Then its differential is found and used to replace the original variable terms. The limits of integration are also changed by plugging the original endpoints into the inner function. This keeps the integral written in one variable from start to finish.

After the change of variables, the integral usually becomes simpler to solve. Because the updated limits are used directly, there is no need to convert back to the original variable at the end. This makes the final numerical answer easier to obtain.

Substitution is useful when standard integration would be difficult or slow. It is also important in applied science and engineering. For example, in electrical engineering, it can help find the total electric charge that flows through a circuit over a time interval by integrating the current. If the current is given by a complicated expression, substitution makes the calculation more practical.

Using Substitution to Evaluate Definite Integrals

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key terms and concepts

Learn how substitution simplifies definite integrals by changing variables, updating limits, and solving the integral directly. Includes a circuit current example.

Using Substitution to Evaluate Definite Integrals | JoVE K12

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Using Substitution to Evaluate Definite Integrals

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