integral of (a+bx^3)^3

Lösung

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

a^{3} x + \frac{3 a ^{2} b x ^{4}}{4} + \frac{3 a b ^{2} x ^{7}}{7} + \frac{b ^{3} x ^{10}}{10} + C

Schritte zur Lösung

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

Multipliziere aus

(a + b x^{3})^{3} : a^{3} + 3 a^{2} b x^{3} + 3 a b^{2} x^{6} + b^{3} x^{9}

= \int a^{3} + 3 a^{2} b x^{3} + 3 a b^{2} x^{6} + b^{3} x^{9} 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} d x + \int 3 a^{2} b x^{3} d x + \int 3 a b^{2} x^{6} d x + \int b^{3} x^{9} d x

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

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

\int 3 a b^{2} x^{6} d x = \frac{3 a b ^{2} x ^{7}}{7}

\int b^{3} x^{9} d x = \frac{b ^{3} x ^{10}}{10}

= a^{3} x + \frac{3 a ^{2} b x ^{4}}{4} + \frac{3 a b ^{2} x ^{7}}{7} + \frac{b ^{3} x ^{10}}{10}

Füge eine Konstante zur Lösung hinzu

= a^{3} x + \frac{3 a ^{2} b x ^{4}}{4} + \frac{3 a b ^{2} x ^{7}}{7} + \frac{b ^{3} x ^{10}}{10} + C

x \to 3 lim (- 2 x^{3})

d er i v a t i v e \frac{1}{x + 4}

\int \frac{20}{( x - 1 ) ( x ^{2} + 9 )} d x

\frac{d}{d x} (\frac{2}{x ^{2} + 1})

d er i v a t i v e (x^{2} + 2 x) (1 - e^{x})