integral ((x^2-a)^3)/2

Lösung

integral

\frac{( x ^{2} - a ) ^{3}}{2}

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Multipliziere aus

(x^{2} - a)^{3} : x^{6} - 3 a x^{4} + 3 a^{2} x^{2} - a^{3}

= \frac{1}{2} \cdot \int x^{6} - 3 a x^{4} + 3 a^{2} x^{2} - a^{3} d x

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 x^{6} d x - \int 3 a x^{4} d x + \int 3 a^{2} x^{2} d x - \int a^{3} d x)

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

\int 3 a x^{4} d x = \frac{3 a x ^{5}}{5}

\int 3 a^{2} x^{2} d x = a^{2} x^{3}

\int a^{3} d x = a^{3} x

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l t (1 - t)

in t e g r a l \frac{x}{( x ^{2} + 1 )}

in t e g r a l x^{2} (sin (x))^{2}

in t e g r a l \frac{y - 2}{x}

in t e g r a l \frac{x ( 1 + 3 x ^{2} ) ^{5}}{2}