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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

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