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受欢迎的微积分 >

integral of sqrt(10^{9x)}

解答

\int 1 0^{9 x} d x

解答

\frac{5 ^{\frac{9 x}{2}} \cdot 2 ^{\frac{1}{2} (2 + 9 x)}}{9 ln ( 10 )} + C

求解步骤

\int 1 0^{9 x} d x

使用根式运算法则:

n a^{m} = a^{\frac{m}{n}},

假定

a \geq 0

= \int 1 0^{\frac{9 x}{2}} d x

使用换元积分法

= \int 1 0^{u} \cdot \frac{2}{9} d u

提出常数:

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

= \frac{2}{9} \cdot \int 1 0^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

= \frac{2}{9} \cdot \frac{1 0 ^{u}}{ln ( 10 )}

u = \frac{9 x}{2}

代回

= \frac{2}{9} \cdot \frac{1 0 ^{\frac{9 x}{2}}}{ln ( 10 )}

化简

\frac{2}{9} \cdot \frac{1 0 ^{\frac{9 x}{2}}}{ln ( 10 )} : \frac{5 ^{\frac{9 x}{2}} \cdot 2 ^{\frac{1}{2} (2 + 9 x)}}{9 ln ( 10 )}

= \frac{5 ^{\frac{9 x}{2}} \cdot 2 ^{\frac{1}{2} (2 + 9 x)}}{9 ln ( 10 )}

解答补常数

= \frac{5 ^{\frac{9 x}{2}} \cdot 2 ^{\frac{1}{2} (2 + 9 x)}}{9 ln ( 10 )} + C

作图

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