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Beliebt Rechnen >

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

Lösung

\int x^{3} (\frac{2 x ^{2} + 1}{x}) d x

Lösung

- 2 x^{5} + \frac{12 x ^{5}}{5} + \frac{x ^{3}}{3} + C

Schritte zur Lösung

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

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

= \int x^{3} (\frac{2 x ^{2}}{x} + \frac{1}{x}) d x

Streiche die gemeinsamen Faktoren:

x

= \int x^{3} (2 x + \frac{1}{x}) d x

Wende die partielle Integration an

= x^{4} (2 x + \frac{1}{x}) - \int x (8 x^{3} + 2 x) d x

\int x (8 x^{3} + 2 x) d x = 4 x^{5} + x^{3} - \frac{12 x ^{5}}{5} - \frac{x ^{3}}{3}

= x^{4} (2 x + \frac{1}{x}) - (4 x^{5} + x^{3} - \frac{12 x ^{5}}{5} - \frac{x ^{3}}{3})

Vereinfache

x^{4} (2 x + \frac{1}{x}) - (4 x^{5} + x^{3} - \frac{12 x ^{5}}{5} - \frac{x ^{3}}{3}) : - 2 x^{5} + \frac{12 x ^{5}}{5} + \frac{x ^{3}}{3}

= - 2 x^{5} + \frac{12 x ^{5}}{5} + \frac{x ^{3}}{3}

Füge eine Konstante zur Lösung hinzu

= - 2 x^{5} + \frac{12 x ^{5}}{5} + \frac{x ^{3}}{3} + C

Graph

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