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Beliebt Rechnen >

integral of (3x^5-7x^3)/(x^2)

Lösung

\int \frac{3 x ^{5} - 7 x ^{3}}{x ^{2}} d x

Lösung

\frac{1}{12} (3 x^{2} - 7)^{2} + C

Schritte zur Lösung

\int \frac{3 x ^{5} - 7 x ^{3}}{x ^{2}} d x

Vereinfache

\frac{3 x ^{5} - 7 x ^{3}}{x ^{2}} : x (3 x^{2} - 7)

= \int x (3 x^{2} - 7) d x

Wende U-Substitution an

= \int \frac{u}{6} d u

Entferne die Konstante:

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

= \frac{1}{6} \cdot \int u d u

Wende die Potenzregel an

= \frac{1}{6} \cdot \frac{u ^{2}}{2}

Setze in

u = 3 x^{2} - 7

ein

= \frac{1}{6} \cdot \frac{( 3 x ^{2} - 7 ) ^{2}}{2}

Vereinfache

\frac{1}{6} \cdot \frac{( 3 x ^{2} - 7 ) ^{2}}{2} : \frac{1}{12} (3 x^{2} - 7)^{2}

= \frac{1}{12} (3 x^{2} - 7)^{2}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{12} (3 x^{2} - 7)^{2} + C

Graph

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