integral of (x^4+x^3)/(x^2+1)

Lösung

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

\frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x - \frac{1}{2} ln x^{2} + 1 + arctan (x) + C

Schritte zur Lösung

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

Lange Division

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

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int x^{2} d x = \frac{x ^{3}}{3}

\int x d x = \frac{x ^{2}}{2}

\int 1 d x = x

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

= \frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x - \frac{1}{2} ln x^{2} + 1 + arctan (x)

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x - \frac{1}{2} ln x^{2} + 1 + arctan (x) + C

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