integral of x(ax^2+b)^3

Lösung

\int x (a x^{2} + b)^{3} d x

\frac{a ^{3} x ^{8}}{8} + \frac{a ^{2} b x ^{6}}{2} + \frac{3 a b ^{2} x ^{4}}{4} + \frac{b ^{3} x ^{2}}{2} + C

Schritte zur Lösung

\int x (a x^{2} + b)^{3} d x

Multipliziere aus

x (a x^{2} + b)^{3} : a^{3} x^{7} + 3 a^{2} b x^{5} + 3 a b^{2} x^{3} + b^{3} x

= \int a^{3} x^{7} + 3 a^{2} b x^{5} + 3 a b^{2} x^{3} + b^{3} x 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

= \int a^{3} x^{7} d x + \int 3 a^{2} b x^{5} d x + \int 3 a b^{2} x^{3} d x + \int b^{3} x d x

\int a^{3} x^{7} d x = \frac{a ^{3} x ^{8}}{8}

\int 3 a^{2} b x^{5} d x = \frac{a ^{2} b x ^{6}}{2}

\int 3 a b^{2} x^{3} d x = \frac{3 a b ^{2} x ^{4}}{4}

\int b^{3} x d x = \frac{b ^{3} x ^{2}}{2}

= \frac{a ^{3} x ^{8}}{8} + \frac{a ^{2} b x ^{6}}{2} + \frac{3 a b ^{2} x ^{4}}{4} + \frac{b ^{3} x ^{2}}{2}

Füge eine Konstante zur Lösung hinzu

= \frac{a ^{3} x ^{8}}{8} + \frac{a ^{2} b x ^{6}}{2} + \frac{3 a b ^{2} x ^{4}}{4} + \frac{b ^{3} x ^{2}}{2} + C

\int - π sin (π x) d x

\frac{d}{d x} (\frac{5}{6 x})

\frac{d}{d y} (1)

l a pl a ce t r an s f or m {e^{t} sin (3 t)}

n = 1 \sum \infty \frac{ln ( 6 n )}{n}