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Beliebt Rechnen >

integral of (4t^2+3t-1)/(t^3-t^2)

Lösung

\int \frac{4 t ^{2} + 3 t - 1}{t ^{3} - t ^{2}} d t

Lösung

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

Schritte zur Lösung

\int \frac{4 t ^{2} + 3 t - 1}{t ^{3} - t ^{2}} d t

Multipliziere aus

\frac{4 t ^{2} + 3 t - 1}{t ^{3} - t ^{2}} : \frac{4 t ^{2}}{t ^{3} - t ^{2}} + \frac{3 t}{t ^{3} - t ^{2}} - \frac{1}{t ^{3} - t ^{2}}

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

\int \frac{4 t ^{2}}{t ^{3} - t ^{2}} d t = 4 ln ∣ t - 1 ∣

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

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

= 4 ln ∣ t - 1 ∣ + 3 (- ln ∣ t ∣ + ln ∣ t - 1 ∣) - (- ln ∣ t ∣ + \frac{1}{t} + ln ∣ t - 1 ∣)

Vereinfache

4 ln ∣ t - 1 ∣ + 3 (- ln ∣ t ∣ + ln ∣ t - 1 ∣) - (- ln ∣ t ∣ + \frac{1}{t} + ln ∣ t - 1 ∣) : 6 ln ∣ t - 1 ∣ - 2 ln ∣ t ∣ - \frac{1}{t}

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (x-arcsin(x))/(sqrt(1-x^2))

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

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

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

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

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

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

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

integral of (4-5x)^{-3/2}

\int (4 - 5 x)^{- \frac{3}{2}} d x