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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

\int \frac{x}{1 + x ^{2}} d x = \frac{1}{2} ln 1 + x^{2}

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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