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Beliebt Rechnen >

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

Lösung

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

Lösung

ln ∣ x ∣ + 2 arctan (x) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ + 2 arctan (x)

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ + 2 arctan (x) + C

Graph

Beliebte Beispiele

integral of (1/(sqrt(x+1)))

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

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\int cot (z) ln (sin (z)) d z

integral of 1/((4x^2-4x+27)^{3/2)}

\int \frac{1}{( 4 x ^{2} - 4 x + 27 ) ^{\frac{3}{2}}} d x

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

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