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

Lösung

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

\frac{1}{2} arctan (x^{2}) + \frac{1}{4} ln 1 + x^{4} + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x + x ^{3}}{1 + x ^{4}} : \frac{x}{1 + x ^{4}} + \frac{x ^{3}}{1 + x ^{4}}

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

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

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

= \frac{1}{2} arctan (x^{2}) + \frac{1}{4} ln 1 + x^{4}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} arctan (x^{2}) + \frac{1}{4} ln 1 + x^{4} + C

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