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Beliebt Rechnen >

integral of 1/(t+t^3)

Lösung

\int \frac{1}{t + t ^{3}} d t

Lösung

ln ∣ t ∣ - \frac{1}{2} ln t^{2} + 1 + C

Schritte zur Lösung

\int \frac{1}{t + t ^{3}} d t

Ermittle den Partialbruch von

\frac{1}{t + t ^{3}} : \frac{1}{t} - \frac{t}{t ^{2} + 1}

= \int \frac{1}{t} - \frac{t}{t ^{2} + 1} d t

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{t} d t - \int \frac{t}{t ^{2} + 1} d t

\int \frac{1}{t} d t = ln ∣ t ∣

\int \frac{t}{t ^{2} + 1} d t = \frac{1}{2} ln t^{2} + 1

= ln ∣ t ∣ - \frac{1}{2} ln t^{2} + 1

Füge eine Konstante zur Lösung hinzu

= ln ∣ t ∣ - \frac{1}{2} ln t^{2} + 1 + C

Graph

Beliebte Beispiele

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\int ln (e^{2 t - 1}) d t

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\int arctan (a) x d x

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\int x^{6} (2 - x) d x

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\int \frac{x ^{2} + 3}{x ^{6} ( x ^{2} + 1 )} d x

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