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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{1}{2} ln x^{2} + 1 + ln x^{2} + 2 + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{3}}{x ^{4} + 3 x ^{2} + 2} : - \frac{x}{x ^{2} + 1} + \frac{2 x}{x ^{2} + 2}

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

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

\int \frac{2 x}{x ^{2} + 2} d x = ln x^{2} + 2

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral from 1 to 7 of 1/(36+(x+5)^2)

\int_{1}^{7} \frac{1}{36 + ( x + 5 ) ^{2}} d x

derivative of (x^2-4/(x^2+4))

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

integral of (-2x^4-4x^2-5)

\int (- 2 x^{4} - 4 x^{2} - 5) d x

integral of 2cos(x/2)

\int 2 cos (\frac{x}{2}) d x

(\partial)/(\partial y)(2ln(x)+ln(y))

\frac{\partial}{\partial y} (2 ln (x) + ln (y))