integral of p(p+3)^6

Lösung

\int p (p + 3)^{6} d p

\frac{p ^{8}}{8} + \frac{18 p ^{7}}{7} + \frac{45 p ^{6}}{2} + 108 p^{5} + \frac{1215 p ^{4}}{4} + 486 p^{3} + \frac{729 p ^{2}}{2} + C

Schritte zur Lösung

\int p (p + 3)^{6} d p

Multipliziere aus

p (p + 3)^{6} : p^{7} + 18 p^{6} + 135 p^{5} + 540 p^{4} + 1215 p^{3} + 1458 p^{2} + 729 p

= \int p^{7} + 18 p^{6} + 135 p^{5} + 540 p^{4} + 1215 p^{3} + 1458 p^{2} + 729 p d p

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int p^{7} d p + \int 18 p^{6} d p + \int 135 p^{5} d p + \int 540 p^{4} d p + \int 1215 p^{3} d p + \int 1458 p^{2} d p + \int 729 p d p

\int p^{7} d p = \frac{p ^{8}}{8}

\int 18 p^{6} d p = \frac{18 p ^{7}}{7}

\int 135 p^{5} d p = \frac{45 p ^{6}}{2}

\int 540 p^{4} d p = 108 p^{5}

\int 1215 p^{3} d p = \frac{1215 p ^{4}}{4}

\int 1458 p^{2} d p = 486 p^{3}

\int 729 p d p = \frac{729 p ^{2}}{2}

= \frac{p ^{8}}{8} + \frac{18 p ^{7}}{7} + \frac{45 p ^{6}}{2} + 108 p^{5} + \frac{1215 p ^{4}}{4} + 486 p^{3} + \frac{729 p ^{2}}{2}

Füge eine Konstante zur Lösung hinzu

= \frac{p ^{8}}{8} + \frac{18 p ^{7}}{7} + \frac{45 p ^{6}}{2} + 108 p^{5} + \frac{1215 p ^{4}}{4} + 486 p^{3} + \frac{729 p ^{2}}{2} + C

x \to 4 lim (x - 4)

x \to 5 lim (25 - x^{2})

y^{^{'}} = 2 y - 4

\frac{\partial}{\partial x} (z^{3} cos (y x))

\frac{d ^{3}}{d x ^{3}} (3 x^{5} + x^{4})