integral of x(1+x^2)arccot(x)

Lösung

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

\frac{1}{4} x^{4} arccot (x) + \frac{x ^{3}}{12} + \frac{1}{2} x^{2} arccot (x) + \frac{1}{4} x - \frac{1}{4} arctan (x) + C

Schritte zur Lösung

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

Multipliziere aus

x (1 + x^{2}) arccot (x) : x arccot (x) + x^{3} arccot (x)

= \int x arccot (x) + x^{3} arccot (x) 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 arccot (x) d x + \int x^{3} arccot (x) d x

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

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

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

Vereinfache

\frac{1}{2} x^{2} arccot (x) + \frac{1}{2} (- arctan (x) + x) + \frac{1}{4} x^{4} arccot (x) + \frac{1}{4} (\frac{x ^{3}}{3} - x + arctan (x)) : \frac{1}{4} x^{4} arccot (x) + \frac{x ^{3}}{12} + \frac{1}{2} x^{2} arccot (x) + \frac{1}{4} x - \frac{1}{4} arctan (x)

= \frac{1}{4} x^{4} arccot (x) + \frac{x ^{3}}{12} + \frac{1}{2} x^{2} arccot (x) + \frac{1}{4} x - \frac{1}{4} arctan (x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} x^{4} arccot (x) + \frac{x ^{3}}{12} + \frac{1}{2} x^{2} arccot (x) + \frac{1}{4} x - \frac{1}{4} arctan (x) + C

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