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Beliebt Rechnen >

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

Lösung

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

Lösung

ln ∣ x ∣ + \frac{1}{x} + 2 ln ∣ x + 1 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

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

Vereinfache

= ln ∣ x ∣ + \frac{1}{x} + 2 ln ∣ x + 1 ∣

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 2x+3x^{1.7}

\int 2 x + 3 x^{1.7} d x

integral of-6+10x^3-8x^5+6x^7

\int - 6 + 10 x^{3} - 8 x^{5} + 6 x^{7} d x

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

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

integral of 1/((u-1)^3)

\int \frac{1}{( u - 1 ) ^{3}} d u

integral of 4x^2-2x

\int 4 x^{2} - 2 x d x