What is the integral of p(p+3)^6 ?

The integral of p(p+3)^6 is (p^8)/8+(18p^7)/7+(45p^6)/2+108p^5+(1215p^4)/4+486p^3+(729p^2)/2+C

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integral of p(p+3)^6

\int p (p + 3)^{6} d p

\frac{p ^{8}}{8} + \frac{18 p ^{7}}{7} + \frac{45 p ^{6}}{2} + 108 p^{5} + \frac{1215 p ^{4}}{4} + 486 p^{3} + \frac{729 p ^{2}}{2} + C

Solution steps

\int p (p + 3)^{6} d p

Expand

p (p + 3)^{6} : p^{7} + 18 p^{6} + 135 p^{5} + 540 p^{4} + 1215 p^{3} + 1458 p^{2} + 729 p

= \int p^{7} + 18 p^{6} + 135 p^{5} + 540 p^{4} + 1215 p^{3} + 1458 p^{2} + 729 p d p

Apply the Sum Rule:

\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 18 p^{6} d p + \int 135 p^{5} d p + \int 540 p^{4} d p + \int 1215 p^{3} d p + \int 1458 p^{2} d p + \int 729 p d p

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

\int 18 p^{6} d p = \frac{18 p ^{7}}{7}

\int 135 p^{5} d p = \frac{45 p ^{6}}{2}

\int 540 p^{4} d p = 108 p^{5}

\int 1215 p^{3} d p = \frac{1215 p ^{4}}{4}

\int 1458 p^{2} d p = 486 p^{3}

\int 729 p d p = \frac{729 p ^{2}}{2}

= \frac{p ^{8}}{8} + \frac{18 p ^{7}}{7} + \frac{45 p ^{6}}{2} + 108 p^{5} + \frac{1215 p ^{4}}{4} + 486 p^{3} + \frac{729 p ^{2}}{2}

Add a constant to the solution

= \frac{p ^{8}}{8} + \frac{18 p ^{7}}{7} + \frac{45 p ^{6}}{2} + 108 p^{5} + \frac{1215 p ^{4}}{4} + 486 p^{3} + \frac{729 p ^{2}}{2} + C

What is the integral of p(p+3)^6 ?
The integral of p(p+3)^6 is (p^8)/8+(18p^7)/7+(45p^6)/2+108p^5+(1215p^4)/4+486p^3+(729p^2)/2+C