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Beliebt Rechnen >

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

Lösung

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

Lösung

t^{2} - t + 5 arctan (t) + C

Schritte zur Lösung

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

Lange Division

\frac{2 t ^{3} - t ^{2} + 2 t + 4}{1 + t ^{2}} : 2 t - 1 + \frac{5}{t ^{2} + 1}

= \int 2 t - 1 + \frac{5}{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

= \int 2 t d t - \int 1 d t + \int \frac{5}{t ^{2} + 1} d t

\int 2 t d t = t^{2}

\int 1 d t = t

\int \frac{5}{t ^{2} + 1} d t = 5 arctan (t)

= t^{2} - t + 5 arctan (t)

Füge eine Konstante zur Lösung hinzu

= t^{2} - t + 5 arctan (t) + C

Graph

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