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人気のある微分積分 >

integral of x^35^{-x}

解

\int x^{3} 5^{- x} d x

解

- \frac{5 ^{- x} x ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{- x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (- \frac{5 ^{- x} x}{ln ( 5 )} - \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- x})) + C

解答ステップ

\int x^{3} \cdot 5^{- x} d x

u置換積分法を適用する

= \int 5^{u} u^{3} d u

部分積分を適用する

= \frac{5 ^{u} u ^{3}}{ln ( 5 )} - \int \frac{3 \cdot 5 ^{u} u ^{2}}{ln ( 5 )} d u

\int \frac{3 \cdot 5 ^{u} u ^{2}}{ln ( 5 )} d u = \frac{3}{ln ( 5 )} (\frac{5 ^{u} u ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{u} u}{ln ( 5 )} - \frac{5 ^{u}}{ln ^{2} ( 5 )}))

= \frac{5 ^{u} u ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{u} u ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{u} u}{ln ( 5 )} - \frac{5 ^{u}}{ln ^{2} ( 5 )}))

代用を戻す

u = - x

= \frac{5 ^{- x} ( - x ) ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{- x} ( - x ) ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{- x} ( - x )}{ln ( 5 )} - \frac{5 ^{- x}}{ln ^{2} ( 5 )}))

簡素化

\frac{5 ^{- x} ( - x ) ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{- x} ( - x ) ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{- x} ( - x )}{ln ( 5 )} - \frac{5 ^{- x}}{ln ^{2} ( 5 )})) : - \frac{5 ^{- x} x ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{- x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (- \frac{5 ^{- x} x}{ln ( 5 )} - \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- x}))

= - \frac{5 ^{- x} x ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{- x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (- \frac{5 ^{- x} x}{ln ( 5 )} - \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- x}))

定数を解答に追加する

= - \frac{5 ^{- x} x ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{- x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (- \frac{5 ^{- x} x}{ln ( 5 )} - \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- x})) + C

グラフ

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