integral of p(p+1)^5

解答

\int p (p + 1)^{5} d p

\frac{p ^{7}}{7} + \frac{5 p ^{6}}{6} + 2 p^{5} + \frac{5 p ^{4}}{2} + \frac{5 p ^{3}}{3} + \frac{p ^{2}}{2} + C

求解步骤

\int p (p + 1)^{5} d p

乘开

p (p + 1)^{5} : p^{6} + 5 p^{5} + 10 p^{4} + 10 p^{3} + 5 p^{2} + p

= \int p^{6} + 5 p^{5} + 10 p^{4} + 10 p^{3} + 5 p^{2} + 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^{6} d p + \int 5 p^{5} d p + \int 10 p^{4} d p + \int 10 p^{3} d p + \int 5 p^{2} d p + \int p d p

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

\int 5 p^{5} d p = \frac{5 p ^{6}}{6}

\int 10 p^{4} d p = 2 p^{5}

\int 10 p^{3} d p = \frac{5 p ^{4}}{2}

\int 5 p^{2} d p = \frac{5 p ^{3}}{3}

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

= \frac{p ^{7}}{7} + \frac{5 p ^{6}}{6} + 2 p^{5} + \frac{5 p ^{4}}{2} + \frac{5 p ^{3}}{3} + \frac{p ^{2}}{2}

解答补常数

= \frac{p ^{7}}{7} + \frac{5 p ^{6}}{6} + 2 p^{5} + \frac{5 p ^{4}}{2} + \frac{5 p ^{3}}{3} + \frac{p ^{2}}{2} + C

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