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Beliebt Rechnen >

integral of (x^4-xarctan(x))/((1+x^2)^2)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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