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Beliebt Rechnen >

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

Lösung

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

Lösung

- ln ∣ x ∣ - \frac{1}{x} + ln ∣ x + 1 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

= - ln ∣ x ∣ - \frac{1}{x} + ln ∣ x + 1 ∣

Füge eine Konstante zur Lösung hinzu

= - ln ∣ x ∣ - \frac{1}{x} + ln ∣ x + 1 ∣ + C

Graph

Beliebte Beispiele

integral of e^{2x}(x^2+3x-1)

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

integral of (sec^4(x))/(tan^4(x))

\int \frac{sec ^{4} ( x )}{tan ^{4} ( x )} d x

integral of x/(1+x^6)

\int \frac{x}{1 + x ^{6}} d x

integral of-9x^3e^{x^4}

\int - 9 x^{3} e^{x^{4}} d x

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

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