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Beliebt Rechnen >

integral of (7(x^2+4))/(x(x^2+7))

Lösung

\int \frac{7 ( x ^{2} + 4 )}{x ( x ^{2} + 7 )} d x

Lösung

\frac{3}{2} ln x^{2} + 7 + 4 ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{7 ( x ^{2} + 4 )}{x ( x ^{2} + 7 )} d x

Entferne die Konstante:

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

= 7 \cdot \int \frac{x ^{2} + 4}{x ( x ^{2} + 7 )} d x

Multipliziere aus

\frac{x ^{2} + 4}{x ( x ^{2} + 7 )} : \frac{x ^{2}}{x ( x ^{2} + 7 )} + \frac{4}{x ( x ^{2} + 7 )}

Wende die Summenregel an:

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

= 7 (\int \frac{x ^{2}}{x ( x ^{2} + 7 )} d x + \int \frac{4}{x ( x ^{2} + 7 )} d x)

\int \frac{x ^{2}}{x ( x ^{2} + 7 )} d x = \frac{1}{2} ln x^{2} + 7

\int \frac{4}{x ( x ^{2} + 7 )} d x = 4 (\frac{1}{7} ln ∣ x ∣ - \frac{1}{14} ln x^{2} + 7)

= 7 (\frac{1}{2} ln x^{2} + 7 + 4 (\frac{1}{7} ln ∣ x ∣ - \frac{1}{14} ln x^{2} + 7))

Vereinfache

7 (\frac{1}{2} ln x^{2} + 7 + 4 (\frac{1}{7} ln ∣ x ∣ - \frac{1}{14} ln x^{2} + 7)) : \frac{3}{2} ln x^{2} + 7 + 4 ln ∣ x ∣

= \frac{3}{2} ln x^{2} + 7 + 4 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{3}{2} ln x^{2} + 7 + 4 ln ∣ x ∣ + C

Graph

Beliebte Beispiele

limit as t approaches 0 of sin(4t)

t \to 0 lim (sin (4 t))

derivative of (x-y+2^2)

\frac{d}{d x} ((x - y + 2)^{2})

(\partial)/(\partial z)(x^2y^4+z^3)

\frac{\partial}{\partial z} (x^{2} y^{4} + z^{3})

integral of x/(x^2-4)

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

y^'-ty^3=0,y(0)=2

y^{^{'}} - t y^{3} = 0, y (0) = 2