integral of p(p+3)^8

解答

\int p (p + 3)^{8} d p

\frac{( p + 3 ) ^{10}}{10} - \frac{( p + 3 ) ^{9}}{3} + C

求解步骤

\int p (p + 3)^{8} d p

使用换元积分法

= \int (u - 3) u^{8} d u

乘开

(u - 3) u^{8} : u^{9} - 3 u^{8}

= \int u^{9} - 3 u^{8} d u

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 u^{9} d u - \int 3 u^{8} d u

\int u^{9} d u = \frac{u ^{10}}{10}

\int 3 u^{8} d u = \frac{u ^{9}}{3}

= \frac{u ^{10}}{10} - \frac{u ^{9}}{3}

u = p + 3

代回

= \frac{( p + 3 ) ^{10}}{10} - \frac{( p + 3 ) ^{9}}{3}

解答补常数

= \frac{( p + 3 ) ^{10}}{10} - \frac{( p + 3 ) ^{9}}{3} + C

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