integral of p(p+2)^7

해법

\int p (p + 2)^{7} d p

\frac{p ^{9}}{9} + \frac{7 p ^{8}}{4} + 12 p^{7} + \frac{140 p ^{6}}{3} + 112 p^{5} + 168 p^{4} + \frac{448 p ^{3}}{3} + 64 p^{2} + C

솔루션 단계

\int p (p + 2)^{7} d p

p (p + 2)^{7}

확대한다:

p^{8} + 14 p^{7} + 84 p^{6} + 280 p^{5} + 560 p^{4} + 672 p^{3} + 448 p^{2} + 128 p

= \int p^{8} + 14 p^{7} + 84 p^{6} + 280 p^{5} + 560 p^{4} + 672 p^{3} + 448 p^{2} + 128 p d p

합계 규칙 적용:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int p^{8} d p + \int 14 p^{7} d p + \int 84 p^{6} d p + \int 280 p^{5} d p + \int 560 p^{4} d p + \int 672 p^{3} d p + \int 448 p^{2} d p + \int 128 p d p

\int p^{8} d p = \frac{p ^{9}}{9}

\int 14 p^{7} d p = \frac{7 p ^{8}}{4}

\int 84 p^{6} d p = 12 p^{7}

\int 280 p^{5} d p = \frac{140 p ^{6}}{3}

\int 560 p^{4} d p = 112 p^{5}

\int 672 p^{3} d p = 168 p^{4}

\int 448 p^{2} d p = \frac{448 p ^{3}}{3}

\int 128 p d p = 64 p^{2}

= \frac{p ^{9}}{9} + \frac{7 p ^{8}}{4} + 12 p^{7} + \frac{140 p ^{6}}{3} + 112 p^{5} + 168 p^{4} + \frac{448 p ^{3}}{3} + 64 p^{2}

솔루션에 상수 추가

= \frac{p ^{9}}{9} + \frac{7 p ^{8}}{4} + 12 p^{7} + \frac{140 p ^{6}}{3} + 112 p^{5} + 168 p^{4} + \frac{448 p ^{3}}{3} + 64 p^{2} + C