integral of x^3*6^{-x}

解答

\int x^{3} \cdot 6^{- x} d x

- \frac{6 ^{- x} x ^{3}}{ln ( 6 )} - \frac{3}{ln ( 6 )} (\frac{6 ^{- x} x ^{2}}{ln ( 6 )} - \frac{2}{ln ( 6 )} (- \frac{6 ^{- x} x}{ln ( 6 )} - \frac{1}{ln ^{2} ( 6 )} \cdot 6^{- x})) + C

求解步骤

\int x^{3} \cdot 6^{- x} d x

使用换元积分法

= \int 6^{u} u^{3} d u

使用分布积分法

= \frac{6 ^{u} u ^{3}}{ln ( 6 )} - \int \frac{2 ^{u} \cdot 3 ^{u + 1} u ^{2}}{ln ( 6 )} d u

\int \frac{2 ^{u} \cdot 3 ^{u + 1} u ^{2}}{ln ( 6 )} d u = \frac{3}{ln ( 6 )} (\frac{6 ^{u} u ^{2}}{ln ( 6 )} - \frac{2}{ln ( 6 )} (\frac{6 ^{u} u}{ln ( 6 )} - \frac{6 ^{u}}{ln ^{2} ( 6 )}))

= \frac{6 ^{u} u ^{3}}{ln ( 6 )} - \frac{3}{ln ( 6 )} (\frac{6 ^{u} u ^{2}}{ln ( 6 )} - \frac{2}{ln ( 6 )} (\frac{6 ^{u} u}{ln ( 6 )} - \frac{6 ^{u}}{ln ^{2} ( 6 )}))

u = - x

代回

= \frac{6 ^{- x} ( - x ) ^{3}}{ln ( 6 )} - \frac{3}{ln ( 6 )} (\frac{6 ^{- x} ( - x ) ^{2}}{ln ( 6 )} - \frac{2}{ln ( 6 )} (\frac{6 ^{- x} ( - x )}{ln ( 6 )} - \frac{6 ^{- x}}{ln ^{2} ( 6 )}))

化简

\frac{6 ^{- x} ( - x ) ^{3}}{ln ( 6 )} - \frac{3}{ln ( 6 )} (\frac{6 ^{- x} ( - x ) ^{2}}{ln ( 6 )} - \frac{2}{ln ( 6 )} (\frac{6 ^{- x} ( - x )}{ln ( 6 )} - \frac{6 ^{- x}}{ln ^{2} ( 6 )})) : - \frac{6 ^{- x} x ^{3}}{ln ( 6 )} - \frac{3}{ln ( 6 )} (\frac{6 ^{- x} x ^{2}}{ln ( 6 )} - \frac{2}{ln ( 6 )} (- \frac{6 ^{- x} x}{ln ( 6 )} - \frac{1}{ln ^{2} ( 6 )} \cdot 6^{- x}))

= - \frac{6 ^{- x} x ^{3}}{ln ( 6 )} - \frac{3}{ln ( 6 )} (\frac{6 ^{- x} x ^{2}}{ln ( 6 )} - \frac{2}{ln ( 6 )} (- \frac{6 ^{- x} x}{ln ( 6 )} - \frac{1}{ln ^{2} ( 6 )} \cdot 6^{- x}))

解答补常数

= - \frac{6 ^{- x} x ^{3}}{ln ( 6 )} - \frac{3}{ln ( 6 )} (\frac{6 ^{- x} x ^{2}}{ln ( 6 )} - \frac{2}{ln ( 6 )} (- \frac{6 ^{- x} x}{ln ( 6 )} - \frac{1}{ln ^{2} ( 6 )} \cdot 6^{- x})) + C

\int (e^{\frac{x}{2}} - e^{- \frac{x}{2}}) d x

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\int_{0}^{7} (7 - x) d x