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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

\frac{3 x ^{2} + 1}{x ^{2} ( 1 + x ^{2} )} : \frac{3 x ^{2}}{x ^{2} ( 1 + x ^{2} )} + \frac{1}{x ^{2} ( 1 + x ^{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{3 x ^{2}}{x ^{2} ( 1 + x ^{2} )} d x + \int \frac{1}{x ^{2} ( 1 + x ^{2} )} d x

\int \frac{3 x ^{2}}{x ^{2} ( 1 + x ^{2} )} d x = 3 arctan (x)

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

= 3 arctan (x) - \frac{1}{x} - arctan (x)

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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