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

Lösung

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

2 (\frac{1}{3} x^{3} (2 - x^{2}) + \frac{2 x ^{5}}{15}) + C

Schritte zur Lösung

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

x^{2} (4 - 2 x^{2}) = 2 x^{2} (2 - x^{2})

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

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

\int - \frac{2 x ^{4}}{3} d x = - \frac{2 x ^{5}}{15}

= 2 (\frac{1}{3} x^{3} (2 - x^{2}) - (- \frac{2 x ^{5}}{15}))

Vereinfache

= 2 (\frac{1}{3} x^{3} (2 - x^{2}) + \frac{2 x ^{5}}{15})

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{3} x^{3} (2 - x^{2}) + \frac{2 x ^{5}}{15}) + C

d er i v a t i v e f (x) = x (x^{3} - x)

\frac{d}{d x} (\frac{6 x - 3}{5 x + 8})

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\int e^{\frac{5}{2} x} d x