de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

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

Lösung

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

Lösung

ln ∣ t ∣ + \frac{1}{2} ln t^{2} + 4 - \frac{1}{2} arctan (\frac{t}{2}) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{t} d t = ln ∣ t ∣

\int \frac{t - 1}{t ^{2} + 4} d t = \frac{1}{2} ln t^{2} + 4 - \frac{1}{2} arctan (\frac{t}{2})

= ln ∣ t ∣ + \frac{1}{2} ln t^{2} + 4 - \frac{1}{2} arctan (\frac{t}{2})

Füge eine Konstante zur Lösung hinzu

= ln ∣ t ∣ + \frac{1}{2} ln t^{2} + 4 - \frac{1}{2} arctan (\frac{t}{2}) + C

Graph

Beliebte Beispiele

integral of (2x-1)^{-1}

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

integral of sqrt(x^4+3x^7)

\int x^{4} + 3 x^{7} d x

integral of ln(sqrt(y)+3)

\int ln (y + 3) d y

integral of 1/(x^3(1+x^3)^{1/3)}

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

integral of 1/(4e^t+e^{-t)}

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