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Beliebt Rechnen >

integral of (6(x^2+4))/(x(x^2+6))

Lösung

\int \frac{6 ( x ^{2} + 4 )}{x ( x ^{2} + 6 )} d x

Lösung

ln x^{2} + 6 + 4 ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{6 ( x ^{2} + 4 )}{x ( x ^{2} + 6 )} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \int \frac{x ^{2} + 4}{x ( x ^{2} + 6 )} d x

Multipliziere aus

\frac{x ^{2} + 4}{x ( x ^{2} + 6 )} : \frac{x ^{2}}{x ( x ^{2} + 6 )} + \frac{4}{x ( x ^{2} + 6 )}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 6 (\int \frac{x ^{2}}{x ( x ^{2} + 6 )} d x + \int \frac{4}{x ( x ^{2} + 6 )} d x)

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

\int \frac{4}{x ( x ^{2} + 6 )} d x = 4 (\frac{1}{6} ln ∣ x ∣ - \frac{1}{12} ln x^{2} + 6)

= 6 (\frac{1}{2} ln x^{2} + 6 + 4 (\frac{1}{6} ln ∣ x ∣ - \frac{1}{12} ln x^{2} + 6))

Vereinfache

6 (\frac{1}{2} ln x^{2} + 6 + 4 (\frac{1}{6} ln ∣ x ∣ - \frac{1}{12} ln x^{2} + 6)) : ln x^{2} + 6 + 4 ln ∣ x ∣

= ln x^{2} + 6 + 4 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= ln x^{2} + 6 + 4 ln ∣ x ∣ + C

Graph

Beliebte Beispiele

integral of 1/(y^2+6y+10)

\int \frac{1}{y ^{2} + 6 y + 10} d y

integral of (x+1)/(x^3-x^2-6x)

\int \frac{x + 1}{x ^{3} - x ^{2} - 6 x} d x

integral of x^2+\sqrt[4]{x}

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

integral of 3/(sqrt(e^{2a)-4)}

\int \frac{3}{e ^{2 a} - 4} d a

integral of 4/(x^2(2x+3))

\int \frac{4}{x ^{2} ( 2 x + 3 )} d x