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Beliebt Rechnen >

integral of (2-3x^2)^2

Lösung

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

Lösung

\frac{2}{3} (26 x - 26 x^{3} + \frac{9 3 x ^{5}}{5 2}) + C

Schritte zur Lösung

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

Wende integrale Substitution an

= \int \frac{2}{3} (2 - 2 u^{2})^{2} d u

Entferne die Konstante:

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

= \frac{2}{3} \cdot \int (2 - 2 u^{2})^{2} d u

Multipliziere aus

(2 - 2 u^{2})^{2} : 4 - 8 u^{2} + 4 u^{4}

= \frac{2}{3} \cdot \int 4 - 8 u^{2} + 4 u^{4} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 4 d u = 4 u

\int 8 u^{2} d u = \frac{8 u ^{3}}{3}

\int 4 u^{4} d u = \frac{4 u ^{5}}{5}

= \frac{2}{3} (4 u - \frac{8 u ^{3}}{3} + \frac{4 u ^{5}}{5})

Setze in

u = \frac{3}{2} x

ein

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

Vereinfache

\frac{2}{3} 4 \cdot \frac{3}{2} x - \frac{8 ( \frac{3}{2} x ) ^{3}}{3} + \frac{4 ( \frac{3}{2} x ) ^{5}}{5} : \frac{2}{3} (26 x - 26 x^{3} + \frac{9 3 x ^{5}}{5 2})

= \frac{2}{3} (26 x - 26 x^{3} + \frac{9 3 x ^{5}}{5 2})

Füge eine Konstante zur Lösung hinzu

= \frac{2}{3} (26 x - 26 x^{3} + \frac{9 3 x ^{5}}{5 2}) + C

Graph

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