What is the integral of p(p+3)^7 ?

The integral of p(p+3)^7 is 1/72 (p+3)^8(8p-3)+C

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integral of p(p+3)^7

\int p (p + 3)^{7} d p

\frac{1}{72} (p + 3)^{8} (8 p - 3) + C

Solution steps

\int p (p + 3)^{7} d p

Apply u-substitution

: \int \frac{7 u ( 7 u - 3 )}{7} d u

= \int \frac{7 u ( 7 u - 3 )}{7} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{7} \cdot \int 7 u (7 u - 3) d u

Expand

7 u (7 u - 3) : u^{\frac{2}{7}} - 3 7 u

= \frac{1}{7} \cdot \int u^{\frac{2}{7}} - 3 7 u d u

Apply the Sum Rule:

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

= \frac{1}{7} (\int u^{\frac{2}{7}} d u - \int 3 7 u d u)

\int u^{\frac{2}{7}} d u = \frac{7}{9} u^{\frac{9}{7}}

\int 3 7 u d u = \frac{21}{8} u^{\frac{8}{7}}

= \frac{1}{7} (\frac{7}{9} u^{\frac{9}{7}} - \frac{21}{8} u^{\frac{8}{7}})

Substitute back

u = (p + 3)^{7}

= \frac{1}{7} (\frac{7}{9} ((p + 3)^{7})^{\frac{9}{7}} - \frac{21}{8} ((p + 3)^{7})^{\frac{8}{7}})

Simplify

\frac{1}{7} (\frac{7}{9} ((p + 3)^{7})^{\frac{9}{7}} - \frac{21}{8} ((p + 3)^{7})^{\frac{8}{7}}) : \frac{1}{72} (p + 3)^{8} (8 p - 3)

= \frac{1}{72} (p + 3)^{8} (8 p - 3)

Add a constant to the solution

= \frac{1}{72} (p + 3)^{8} (8 p - 3) + C

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What is the integral of p(p+3)^7 ?
The integral of p(p+3)^7 is 1/72 (p+3)^8(8p-3)+C