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Beliebt Rechnen >

integral of ((x^3))/(sqrt(1-2x^2))

Lösung

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

Lösung

- \frac{1}{6} (x^{2} + 1) - 2 x^{2} + 1 + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{u ^{2} - 1}{4} d u

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int u^{2} - 1 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{1}{4} (\int u^{2} d u - \int 1 d u)

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

\int 1 d u = u

= \frac{1}{4} (\frac{u ^{3}}{3} - u)

Setze in

u = 1 - 2 x^{2}

ein

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

Vereinfache

\frac{1}{4} (\frac{( 1 - 2 x ^{2} ) ^{3}}{3} - 1 - 2 x^{2}) : - \frac{1}{6} (x^{2} + 1) - 2 x^{2} + 1

= - \frac{1}{6} (x^{2} + 1) - 2 x^{2} + 1

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{6} (x^{2} + 1) - 2 x^{2} + 1 + C

Graph

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