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

Lösung

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

\frac{1}{2} ln x^{2} + 1 + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Vereinfache

\frac{1}{2} (ln x^{2} + 1 + \frac{1}{x ^{2} + 1}) - \frac{1}{2 ( x ^{2} + 1 )} : \frac{1}{2} ln x^{2} + 1

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

Füge eine Konstante zur Lösung hinzu

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

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