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Beliebt Rechnen >

integral of (3x^2+1)/((x+1)(x-5)^2)

Lösung

\int \frac{3 x ^{2} + 1}{( x + 1 ) ( x - 5 ) ^{2}} d x

Lösung

\frac{1}{9} ln ∣ x + 1 ∣ + \frac{26}{9} ln ∣ x - 5 ∣ - \frac{38}{3 ( x - 5 )} + C

Schritte zur Lösung

\int \frac{3 x ^{2} + 1}{( x + 1 ) ( x - 5 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{3 x ^{2} + 1}{( x + 1 ) ( x - 5 ) ^{2}} : \frac{1}{9 ( x + 1 )} + \frac{26}{9 ( x - 5 )} + \frac{38}{3 ( x - 5 ) ^{2}}

= \int \frac{1}{9 ( x + 1 )} + \frac{26}{9 ( x - 5 )} + \frac{38}{3 ( x - 5 ) ^{2}} 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}{9 ( x + 1 )} d x + \int \frac{26}{9 ( x - 5 )} d x + \int \frac{38}{3 ( x - 5 ) ^{2}} d x

\int \frac{1}{9 ( x + 1 )} d x = \frac{1}{9} ln ∣ x + 1 ∣

\int \frac{26}{9 ( x - 5 )} d x = \frac{26}{9} ln ∣ x - 5 ∣

\int \frac{38}{3 ( x - 5 ) ^{2}} d x = - \frac{38}{3 ( x - 5 )}

= \frac{1}{9} ln ∣ x + 1 ∣ + \frac{26}{9} ln ∣ x - 5 ∣ - \frac{38}{3 ( x - 5 )}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{9} ln ∣ x + 1 ∣ + \frac{26}{9} ln ∣ x - 5 ∣ - \frac{38}{3 ( x - 5 )} + C

Graph

Beliebte Beispiele

integral of x(x-8)^{3/4}

\int x (x - 8)^{\frac{3}{4}} d x

integral of (3-x^2)

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

integral of 2x^7sqrt(x^4+1)

\int 2 x^{7} x^{4} + 1 d x

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

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

integral of (5x^2)/((x-2)(x^2+1))

\int \frac{5 x ^{2}}{( x - 2 ) ( x ^{2} + 1 )} d x