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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

= \frac{( x + 1 ) ^{2}}{2} - 2 (x + 1) + ln ∣ x + 1 ∣ + 4 (ln ∣ x ∣ - ln ∣ x + 1 ∣)

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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

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

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

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

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

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

integral of 1/(g-kv^2)

\int \frac{1}{g - k v ^{2}} d v

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

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