integral of p(p+7)^6

解答

\int p (p + 7)^{6} d p

\frac{p ^{8}}{8} + 6 p^{7} + \frac{245 p ^{6}}{2} + 1372 p^{5} + \frac{36015 p ^{4}}{4} + 33614 p^{3} + \frac{117649 p ^{2}}{2} + C

求解步骤

\int p (p + 7)^{6} d p

乘开

p (p + 7)^{6} : p^{7} + 42 p^{6} + 735 p^{5} + 6860 p^{4} + 36015 p^{3} + 100842 p^{2} + 117649 p

= \int p^{7} + 42 p^{6} + 735 p^{5} + 6860 p^{4} + 36015 p^{3} + 100842 p^{2} + 117649 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 42 p^{6} d p + \int 735 p^{5} d p + \int 6860 p^{4} d p + \int 36015 p^{3} d p + \int 100842 p^{2} d p + \int 117649 p d p

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

\int 42 p^{6} d p = 6 p^{7}

\int 735 p^{5} d p = \frac{245 p ^{6}}{2}

\int 6860 p^{4} d p = 1372 p^{5}

\int 36015 p^{3} d p = \frac{36015 p ^{4}}{4}

\int 100842 p^{2} d p = 33614 p^{3}

\int 117649 p d p = \frac{117649 p ^{2}}{2}

= \frac{p ^{8}}{8} + 6 p^{7} + \frac{245 p ^{6}}{2} + 1372 p^{5} + \frac{36015 p ^{4}}{4} + 33614 p^{3} + \frac{117649 p ^{2}}{2}

解答补常数

= \frac{p ^{8}}{8} + 6 p^{7} + \frac{245 p ^{6}}{2} + 1372 p^{5} + \frac{36015 p ^{4}}{4} + 33614 p^{3} + \frac{117649 p ^{2}}{2} + C

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