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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 1/(x+3+sqrt(x+3))

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

integral of (2r)/((1-r)^{2/3)}

\int \frac{2 r}{( 1 - r ) ^{\frac{2}{3}}} d r

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

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

integral of 6x^2(2x^3+5)^5

\int 6 x^{2} (2 x^{3} + 5)^{5} d x

integral of (2x)/y-(3y^2)/(x^4)

\int \frac{2 x}{y} - \frac{3 y ^{2}}{x ^{4}} d x