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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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

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

integral of (1+x^2)/(2x)

\int \frac{1 + x ^{2}}{2 x} d x

integral of (x^2+x+8)sqrt(x+2)

\int (x^{2} + x + 8) x + 2 d x

integral of (x^2)/(sqrt(12-5x^3))

\int \frac{x ^{2}}{12 - 5 x ^{3}} d x

integral of ((2x^3))/(x^2+x)

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