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Beliebt Rechnen >

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

Lösung

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

Lösung

x^{3} + arctan (x) + C

Schritte zur Lösung

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

Multipliziere aus

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

\int \frac{3 x ^{4}}{x ^{2} + 1} d x = x^{3} - 3 x + 3 arctan (x)

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

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

= x^{3} - 3 x + 3 arctan (x) + 3 (- arctan (x) + x) + arctan (x)

Vereinfache

x^{3} - 3 x + 3 arctan (x) + 3 (- arctan (x) + x) + arctan (x) : x^{3} + arctan (x)

= x^{3} + arctan (x)

Füge eine Konstante zur Lösung hinzu

= x^{3} + arctan (x) + C

Graph

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