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Beliebt Rechnen >

integral of (x^3)/((x^2+12)^3)

Lösung

\int \frac{x ^{3}}{( x ^{2} + 12 ) ^{3}} d x

Lösung

- \frac{1}{2 ( x ^{2} + 12 )} + \frac{3}{( x ^{2} + 12 ) ^{2}} + C

Schritte zur Lösung

\int \frac{x ^{3}}{( x ^{2} + 12 ) ^{3}} d x

Ermittle den Partialbruch von

\frac{x ^{3}}{( x ^{2} + 12 ) ^{3}} : \frac{x}{( x ^{2} + 12 ) ^{2}} - \frac{12 x}{( x ^{2} + 12 ) ^{3}}

= \int \frac{x}{( x ^{2} + 12 ) ^{2}} - \frac{12 x}{( x ^{2} + 12 ) ^{3}} d x

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}{( x ^{2} + 12 ) ^{2}} d x - \int \frac{12 x}{( x ^{2} + 12 ) ^{3}} d x

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

\int \frac{12 x}{( x ^{2} + 12 ) ^{3}} d x = - \frac{3}{( x ^{2} + 12 ) ^{2}}

= - \frac{1}{2 ( x ^{2} + 12 )} - (- \frac{3}{( x ^{2} + 12 ) ^{2}})

Vereinfache

= - \frac{1}{2 ( x ^{2} + 12 )} + \frac{3}{( x ^{2} + 12 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2 ( x ^{2} + 12 )} + \frac{3}{( x ^{2} + 12 ) ^{2}} + C

Graph

Beliebte Beispiele

(dy)/(dx)= 1/(x^2y^2)

\frac{d y}{d x} = \frac{1}{x ^{2} y ^{2}}

maclaurin x^2e^{4x}

ma c l a u r in x^{2} e^{4 x}

(dy)/(dx)=e^{-ka*x},y(0)=0

\frac{d y}{d x} = e^{- ka \cdot x}, y (0) = 0

integral of (x^2-x+12)/(x^3+6x)

\int \frac{x ^{2} - x + 12}{x ^{3} + 6 x} d x

(\partial)/(\partial x)(e^{3x}sin(cy))

\frac{\partial}{\partial x} (e^{3 x} sin (cy))