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受欢迎的微积分 >

integral of \sqrt[3]{4^{2x}}

解答

\int 3 4^{2 x} d x

解答

\frac{1}{ln ( 2 )} (\frac{3 \cdot 4 ^{\frac{8 x}{3}}}{1 6 ^{x + 1}} + \frac{9}{16} \cdot 4^{\frac{2 x}{3}}) + C

求解步骤

\int 3 4^{2 x} d x

使用换元积分法

= \int \frac{3 u}{ln ( 2 ) \cdot 2 ^{\frac{l n ( u ) + 2 l n ( 2 )}{l n ( 2 )}}} d u

提出常数:

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

= \frac{1}{ln ( 2 )} \cdot \int \frac{3 u}{2 ^{\frac{l n ( u ) + 2 l n ( 2 )}{l n ( 2 )}}} d u

使用分布积分法

= \frac{1}{ln ( 2 )} (\frac{3 u ^{\frac{4}{3}}}{2 ^{\frac{l n ( u ) + 4 l n ( 2 )}{l n ( 2 )}}} - \int - \frac{3}{16 u ^{\frac{2}{3}}} d u)

\int - \frac{3}{16 u ^{\frac{2}{3}}} d u = - \frac{9}{16} u^{\frac{1}{3}}

= \frac{1}{ln ( 2 )} (\frac{3 u ^{\frac{4}{3}}}{2 ^{\frac{l n ( u ) + 4 l n ( 2 )}{l n ( 2 )}}} - (- \frac{9}{16} u^{\frac{1}{3}}))

u = 4^{2 x}

代回

= \frac{1}{ln ( 2 )} \frac{3 ( 4 ^{2 x} ) ^{\frac{4}{3}}}{2 ^{\frac{l n ( 4 ^{2 x} ) + 4 l n ( 2 )}{l n ( 2 )}}} - (- \frac{9}{16} (4^{2 x})^{\frac{1}{3}})

化简

\frac{1}{ln ( 2 )} \frac{3 ( 4 ^{2 x} ) ^{\frac{4}{3}}}{2 ^{\frac{l n ( 4 ^{2 x} ) + 4 l n ( 2 )}{l n ( 2 )}}} - (- \frac{9}{16} (4^{2 x})^{\frac{1}{3}}) : \frac{1}{ln ( 2 )} (\frac{3 \cdot 4 ^{\frac{8 x}{3}}}{1 6 ^{x + 1}} + \frac{9}{16} \cdot 4^{\frac{2 x}{3}})

= \frac{1}{ln ( 2 )} (\frac{3 \cdot 4 ^{\frac{8 x}{3}}}{1 6 ^{x + 1}} + \frac{9}{16} \cdot 4^{\frac{2 x}{3}})

解答补常数

= \frac{1}{ln ( 2 )} (\frac{3 \cdot 4 ^{\frac{8 x}{3}}}{1 6 ^{x + 1}} + \frac{9}{16} \cdot 4^{\frac{2 x}{3}}) + C

作图

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