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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

= - \frac{1}{x} - arctan (x) - arctan (x)

Vereinfache

= - \frac{1}{x} - 2 arctan (x)

Füge eine Konstante zur Lösung hinzu

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

\int 8 x e^{x^{2} + 1} d x

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

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