integral of p(p+7)^8

解答

\int p (p + 7)^{8} d p

\frac{( p + 7 ) ^{10}}{10} - \frac{7 ( p + 7 ) ^{9}}{9} + C

求解步骤

\int p (p + 7)^{8} d p

使用换元积分法

= \int (u - 7) u^{8} d u

乘开

(u - 7) u^{8} : u^{9} - 7 u^{8}

= \int u^{9} - 7 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 7 u^{8} d u

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

\int 7 u^{8} d u = \frac{7 u ^{9}}{9}

= \frac{u ^{10}}{10} - \frac{7 u ^{9}}{9}

u = p + 7

代回

= \frac{( p + 7 ) ^{10}}{10} - \frac{7 ( p + 7 ) ^{9}}{9}

解答补常数

= \frac{( p + 7 ) ^{10}}{10} - \frac{7 ( p + 7 ) ^{9}}{9} + C

y^{^{''}} + 2 y^{^{'}} + y = 4 e^{- 1}

\frac{d}{d t} (tan (t + \frac{π}{4}))

\frac{d y}{d x} = y - y^{5}

\frac{d ^{2} y}{d x ^{2}} - 2 \cdot \frac{d y}{d x} + a y = 0, a > 1

d er i v a t i v e \frac{x ^{5} + 1}{x}