integral of x*(x+5)^4

Lösung

\int x \cdot (x + 5)^{4} d x

\frac{x ^{6}}{6} + 4 x^{5} + \frac{75 x ^{4}}{2} + \frac{500 x ^{3}}{3} + \frac{625 x ^{2}}{2} + C

Schritte zur Lösung

\int x (x + 5)^{4} d x

Multipliziere aus

x (x + 5)^{4} : x^{5} + 20 x^{4} + 150 x^{3} + 500 x^{2} + 625 x

= \int x^{5} + 20 x^{4} + 150 x^{3} + 500 x^{2} + 625 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 x^{5} d x + \int 20 x^{4} d x + \int 150 x^{3} d x + \int 500 x^{2} d x + \int 625 x d x

\int x^{5} d x = \frac{x ^{6}}{6}

\int 20 x^{4} d x = 4 x^{5}

\int 150 x^{3} d x = \frac{75 x ^{4}}{2}

\int 500 x^{2} d x = \frac{500 x ^{3}}{3}

\int 625 x d x = \frac{625 x ^{2}}{2}

= \frac{x ^{6}}{6} + 4 x^{5} + \frac{75 x ^{4}}{2} + \frac{500 x ^{3}}{3} + \frac{625 x ^{2}}{2}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{6}}{6} + 4 x^{5} + \frac{75 x ^{4}}{2} + \frac{500 x ^{3}}{3} + \frac{625 x ^{2}}{2} + C

\int (3 x^{2} + 1)^{3} 6 x d x

\int \frac{a ^{2}}{x x ^{2} - a ^{2}} d x

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

a re a x^{6}, [0, 1]

\int \frac{x}{x + 23} d x