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

Lösung

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

\frac{1}{35} x (- 40 x^{6} + 84 x^{4} - 70 x^{2} + 35) + C

Schritte zur Lösung

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

Wende integrale Substitution an

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

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int (1 - u^{2})^{3} d u

Multipliziere aus

(1 - u^{2})^{3} : 1 - 3 u^{2} + 3 u^{4} - u^{6}

= \frac{1}{2} \cdot \int 1 - 3 u^{2} + 3 u^{4} - u^{6} 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}{2} (\int 1 d u - \int 3 u^{2} d u + \int 3 u^{4} d u - \int u^{6} d u)

\int 1 d u = u

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

\int 3 u^{4} d u = \frac{3 u ^{5}}{5}

\int u^{6} d u = \frac{u ^{7}}{7}

= \frac{1}{2} (u - u^{3} + \frac{3 u ^{5}}{5} - \frac{u ^{7}}{7})

Setze in

u = 2 x

ein

= \frac{1}{2} (2 x - (2 x)^{3} + \frac{3 ( 2 x ) ^{5}}{5} - \frac{( 2 x ) ^{7}}{7})

Vereinfache

\frac{1}{2} (2 x - (2 x)^{3} + \frac{3 ( 2 x ) ^{5}}{5} - \frac{( 2 x ) ^{7}}{7}) : \frac{1}{35} x (- 40 x^{6} + 84 x^{4} - 70 x^{2} + 35)

= \frac{1}{35} x (- 40 x^{6} + 84 x^{4} - 70 x^{2} + 35)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{35} x (- 40 x^{6} + 84 x^{4} - 70 x^{2} + 35) + C

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