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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (cos(x))/(x^4)

\int \frac{cos ( x )}{x ^{4}} d x

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

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

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\int x (x^{2} + 6)^{67} d x