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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{1}{2} ln ∣ t ∣ + \frac{1}{3} ln ∣ t - 1 ∣ + \frac{1}{6} ln ∣ t + 2 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{t ^{3} + t ^{2} - 2 t} : - \frac{1}{2 t} + \frac{1}{3 ( t - 1 )} + \frac{1}{6 ( t + 2 )}

= \int - \frac{1}{2 t} + \frac{1}{3 ( t - 1 )} + \frac{1}{6 ( t + 2 )} 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}{2 t} d t + \int \frac{1}{3 ( t - 1 )} d t + \int \frac{1}{6 ( t + 2 )} d t

\int \frac{1}{2 t} d t = \frac{1}{2} ln ∣ t ∣

\int \frac{1}{3 ( t - 1 )} d t = \frac{1}{3} ln ∣ t - 1 ∣

\int \frac{1}{6 ( t + 2 )} d t = \frac{1}{6} ln ∣ t + 2 ∣

= - \frac{1}{2} ln ∣ t ∣ + \frac{1}{3} ln ∣ t - 1 ∣ + \frac{1}{6} ln ∣ t + 2 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln ∣ t ∣ + \frac{1}{3} ln ∣ t - 1 ∣ + \frac{1}{6} ln ∣ t + 2 ∣ + C

Graph

Beliebte Beispiele

integral of (32x^2)/((x-15)(x+5)^2)

\int \frac{32 x ^{2}}{( x - 15 ) ( x + 5 ) ^{2}} d x

integral of 8^{-x}

\int 8^{- x} d x

derivative of 1/(sec^2(4x))

\frac{d}{d x} (\frac{1}{sec ^{2} ( 4 x )})

derivative f(x)=ln(6)-3x^5

d er i v a t i v e f (x) = ln (6) - 3 x^{5}

derivative (x^2)/(x^2+1)

d er i v a t i v e \frac{x ^{2}}{x ^{2} + 1}