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受欢迎的微积分 >

integral of ((sqrt(x)+8)^9)/(2sqrt(x))

解答

\int \frac{( x + 8 ) ^{9}}{2 x} d x

解答

x (x + 8)^{9} - \frac{9}{10} (x + 8)^{10} + 8 (x + 8)^{9} + C

求解步骤

\int \frac{( x + 8 ) ^{9}}{2 x} d x

提出常数:

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

= \frac{1}{2} \cdot \int \frac{( x + 8 ) ^{9}}{x} d x

使用换元积分法

= \frac{1}{2} \cdot \int 2 (u + 8)^{9} d u

提出常数:

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

= \frac{1}{2} \cdot 2 \cdot \int (u + 8)^{9} d u

使用分布积分法

= \frac{1}{2} \cdot 2 ((u + 8)^{9} u - \int 9 (u + 8)^{8} u d u)

\int 9 (u + 8)^{8} u d u = \frac{9}{10} (u + 8)^{10} - 8 (u + 8)^{9}

= \frac{1}{2} \cdot 2 ((u + 8)^{9} u - (\frac{9}{10} (u + 8)^{10} - 8 (u + 8)^{9}))

u = x

代回

= \frac{1}{2} \cdot 2 ((x + 8)^{9} x - (\frac{9}{10} (x + 8)^{10} - 8 (x + 8)^{9}))

化简

\frac{1}{2} \cdot 2 ((x + 8)^{9} x - (\frac{9}{10} (x + 8)^{10} - 8 (x + 8)^{9})) : x (x + 8)^{9} - \frac{9}{10} (x + 8)^{10} + 8 (x + 8)^{9}

= x (x + 8)^{9} - \frac{9}{10} (x + 8)^{10} + 8 (x + 8)^{9}

解答补常数

= x (x + 8)^{9} - \frac{9}{10} (x + 8)^{10} + 8 (x + 8)^{9} + C

作图

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