integral of p(p+6)^6

解

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

\frac{p ^{8}}{8} + \frac{36 p ^{7}}{7} + 90 p^{6} + 864 p^{5} + 4860 p^{4} + 15552 p^{3} + 23328 p^{2} + C

解答ステップ

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

拡張

p (p + 6)^{6} : p^{7} + 36 p^{6} + 540 p^{5} + 4320 p^{4} + 19440 p^{3} + 46656 p^{2} + 46656 p

= \int p^{7} + 36 p^{6} + 540 p^{5} + 4320 p^{4} + 19440 p^{3} + 46656 p^{2} + 46656 p d p

総和規則を適用する:

\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 36 p^{6} d p + \int 540 p^{5} d p + \int 4320 p^{4} d p + \int 19440 p^{3} d p + \int 46656 p^{2} d p + \int 46656 p d p

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

\int 36 p^{6} d p = \frac{36 p ^{7}}{7}

\int 540 p^{5} d p = 90 p^{6}

\int 4320 p^{4} d p = 864 p^{5}

\int 19440 p^{3} d p = 4860 p^{4}

\int 46656 p^{2} d p = 15552 p^{3}

\int 46656 p d p = 23328 p^{2}

= \frac{p ^{8}}{8} + \frac{36 p ^{7}}{7} + 90 p^{6} + 864 p^{5} + 4860 p^{4} + 15552 p^{3} + 23328 p^{2}

定数を解答に追加する

= \frac{p ^{8}}{8} + \frac{36 p ^{7}}{7} + 90 p^{6} + 864 p^{5} + 4860 p^{4} + 15552 p^{3} + 23328 p^{2} + C