integral of p(p+8)^6

解答

\int p (p + 8)^{6} d p

\frac{p ^{8}}{8} + \frac{48 p ^{7}}{7} + 160 p^{6} + 2048 p^{5} + 15360 p^{4} + 65536 p^{3} + 131072 p^{2} + C

求解步骤

\int p (p + 8)^{6} d p

乘开

p (p + 8)^{6} : p^{7} + 48 p^{6} + 960 p^{5} + 10240 p^{4} + 61440 p^{3} + 196608 p^{2} + 262144 p

= \int p^{7} + 48 p^{6} + 960 p^{5} + 10240 p^{4} + 61440 p^{3} + 196608 p^{2} + 262144 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 48 p^{6} d p + \int 960 p^{5} d p + \int 10240 p^{4} d p + \int 61440 p^{3} d p + \int 196608 p^{2} d p + \int 262144 p d p

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

\int 48 p^{6} d p = \frac{48 p ^{7}}{7}

\int 960 p^{5} d p = 160 p^{6}

\int 10240 p^{4} d p = 2048 p^{5}

\int 61440 p^{3} d p = 15360 p^{4}

\int 196608 p^{2} d p = 65536 p^{3}

\int 262144 p d p = 131072 p^{2}

= \frac{p ^{8}}{8} + \frac{48 p ^{7}}{7} + 160 p^{6} + 2048 p^{5} + 15360 p^{4} + 65536 p^{3} + 131072 p^{2}

解答补常数

= \frac{p ^{8}}{8} + \frac{48 p ^{7}}{7} + 160 p^{6} + 2048 p^{5} + 15360 p^{4} + 65536 p^{3} + 131072 p^{2} + C

\int \frac{1 - x ^{8}}{x ( 1 + x ^{8} )} d x

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

\int 2^{- 2 x} d x

\int \frac{10 x ^{2} - 2 x - 6}{x ^{3} - x ^{2} - 2 x} d x

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