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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{2} + 1}{x ( x ^{2} + 3 )} : \frac{x ^{2}}{x ( x ^{2} + 3 )} + \frac{1}{x ( x ^{2} + 3 )}

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

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

\int \frac{1}{x ( x ^{2} + 3 )} d x = \frac{1}{3} ln ∣ x ∣ - \frac{1}{6} ln x^{2} + 3

= \frac{1}{2} ln x^{2} + 3 + \frac{1}{3} ln ∣ x ∣ - \frac{1}{6} ln x^{2} + 3

Addiere gleiche Elemente:

\frac{1}{2} ln x^{2} + 3 - \frac{1}{6} ln x^{2} + 3 = \frac{1}{3} ln x^{2} + 3

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

derivative 1/x cos(x)

d er i v a t i v e \frac{1}{x} cos (x)

inverselaplace (s+5)/(7s^2+26s+25)

in v erse l a pl a ce \frac{s + 5}{7 s ^{2} + 26 s + 25}

\frac{d y}{d t} + 4 y = 0

tangent f(x)=x^7,\at x=2

t an g e n t f (x) = x^{7}, at x = 2

(dv)/(dt)=-ktv^2,v(0)=50

\frac{d v}{d t} = - k t v^{2}, v (0) = 50