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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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\int cos^{6} (4 θ) d θ

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