integral of p(p+6)^6

解答

\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

求解步骤

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

乘开

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

使用积分加法定则:

\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}

解答补常数

= \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

\int 5 \frac{x}{16 - x ^{2}} d x

\int (- 5 x^{4} + 8 x^{3}) (- 2 x^{5} + 4 x^{4})^{6} d x

x \to 2 lim (\frac{( 3 x + 2 )}{5 x - 2})^{\frac{( 3 )}{x - 2}}

x \to \infty lim (x^{6} e^{- x^{5}})

\int sinh (1 + 4 x) d x