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受欢迎的微积分 >

integral of x(6x+7)^8

解答

\int x (6 x + 7)^{8} d x

解答

\frac{1}{3240} (9 (6 x + 7)^{10} - 70 (6 x + 7)^{9}) + C

求解步骤

\int x (6 x + 7)^{8} d x

使用换元积分法

= \int \frac{u ^{8} ( u - 7 )}{36} d u

提出常数:

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

= \frac{1}{36} \cdot \int u^{8} (u - 7) d u

乘开

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

= \frac{1}{36} \cdot \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

= \frac{1}{36} (\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{1}{36} (\frac{u ^{10}}{10} - \frac{7 u ^{9}}{9})

u = 6 x + 7

代回

= \frac{1}{36} (\frac{( 6 x + 7 ) ^{10}}{10} - \frac{7 ( 6 x + 7 ) ^{9}}{9})

化简

\frac{1}{36} (\frac{( 6 x + 7 ) ^{10}}{10} - \frac{7 ( 6 x + 7 ) ^{9}}{9}) : \frac{1}{3240} (9 (6 x + 7)^{10} - 70 (6 x + 7)^{9})

= \frac{1}{3240} (9 (6 x + 7)^{10} - 70 (6 x + 7)^{9})

解答补常数

= \frac{1}{3240} (9 (6 x + 7)^{10} - 70 (6 x + 7)^{9}) + C

作图

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