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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

\int \frac{2 x}{x ^{3} + x ^{2}} d x = 2 (ln ∣ x ∣ - ln ∣ x + 1 ∣)

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

Vereinfache

3 (\frac{x}{x + 1} + ln ∣ x + 1 ∣ + \frac{1}{x + 1}) + 2 (ln ∣ x ∣ - ln ∣ x + 1 ∣) : \frac{3 x}{x + 1} + ln ∣ x + 1 ∣ + \frac{3}{x + 1} + 2 ln ∣ x ∣

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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

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

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

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

integral of xcos^2(x/2)

\int x cos^{2} (\frac{x}{2}) d x

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

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

integral of 2x^2(x^3+8)^5

\int 2 x^{2} (x^{3} + 8)^{5} d x