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Beliebt Rechnen >

integral of 3/(t(1+t^2))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int \frac{1}{t ( 1 + t ^{2} )} d t

Ermittle den Partialbruch von

\frac{1}{t ( 1 + t ^{2} )} : \frac{1}{t} - \frac{t}{t ^{2} + 1}

= 3 \cdot \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

= 3 (\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

= 3 (ln ∣ t ∣ - \frac{1}{2} ln t^{2} + 1)

Füge eine Konstante zur Lösung hinzu

= 3 (ln ∣ t ∣ - \frac{1}{2} ln t^{2} + 1) + C

Graph

Beliebte Beispiele

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\int \frac{4 x ^{2} - x}{x} d x

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\int \frac{1}{g - k v ^{2}} d v

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\int \frac{x ^{2}}{x ^{2} + y ^{2}} d x

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\int \frac{\partial}{\partial x} (x^{2}) d x

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\int tan^{2} (1 - x) d x