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Beliebt Rechnen >

integral of (2x^5)/(3x^2+4)

Lösung

\int \frac{2 x ^{5}}{3 x ^{2} + 4} d x

Lösung

2 (\frac{x ^{4}}{12} - \frac{2 x ^{2}}{9} + \frac{8}{27} ln 3 x^{2} + 4) + C

Schritte zur Lösung

\int \frac{2 x ^{5}}{3 x ^{2} + 4} d x

Entferne die Konstante:

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

= 2 \cdot \int \frac{x ^{5}}{3 x ^{2} + 4} d x

Lange Division

\frac{x ^{5}}{3 x ^{2} + 4} : \frac{x ^{3}}{3} - \frac{4 x}{9} + \frac{16 x}{9 ( 3 x ^{2} + 4 )}

= 2 \cdot \int \frac{x ^{3}}{3} - \frac{4 x}{9} + \frac{16 x}{9 ( 3 x ^{2} + 4 )} 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

= 2 (\int \frac{x ^{3}}{3} d x - \int \frac{4 x}{9} d x + \int \frac{16 x}{9 ( 3 x ^{2} + 4 )} d x)

\int \frac{x ^{3}}{3} d x = \frac{x ^{4}}{12}

\int \frac{4 x}{9} d x = \frac{2 x ^{2}}{9}

\int \frac{16 x}{9 ( 3 x ^{2} + 4 )} d x = \frac{8}{27} ln 3 x^{2} + 4

= 2 (\frac{x ^{4}}{12} - \frac{2 x ^{2}}{9} + \frac{8}{27} ln 3 x^{2} + 4)

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{x ^{4}}{12} - \frac{2 x ^{2}}{9} + \frac{8}{27} ln 3 x^{2} + 4) + C

Graph

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(dy)/(dx)=0.276y-0.0012y^2