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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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