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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{2} ln ∣ x + 1 ∣ + \frac{3}{4} ln ∣ x - 1 ∣ - \frac{1}{4} ln ∣ x + 3 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{2} + 4 x + 1}{( x ^{2} - 1 ) ( x + 3 )} : \frac{1}{2 ( x + 1 )} + \frac{3}{4 ( x - 1 )} - \frac{1}{4 ( x + 3 )}

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

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

\int \frac{3}{4 ( x - 1 )} d x = \frac{3}{4} ln ∣ x - 1 ∣

\int \frac{1}{4 ( x + 3 )} d x = \frac{1}{4} ln ∣ x + 3 ∣

= \frac{1}{2} ln ∣ x + 1 ∣ + \frac{3}{4} ln ∣ x - 1 ∣ - \frac{1}{4} ln ∣ x + 3 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln ∣ x + 1 ∣ + \frac{3}{4} ln ∣ x - 1 ∣ - \frac{1}{4} ln ∣ x + 3 ∣ + C

Graph

Beliebte Beispiele

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

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

integral of (1/(1+x^4))

\int (\frac{1}{1 + x ^{4}}) d x

integral of-3ln(x)

\int - 3 ln (x) d x

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

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

integral of (e^xsin(e^x))

\int (e^{x} sin (e^{x})) d x