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人気のある微分積分 >

integral of x^24^{-x}

解

\int x^{2} 4^{- x} d x

解

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

解答ステップ

\int x^{2} \cdot 4^{- x} d x

u置換積分法を適用する

= \int - 4^{u} u^{2} d u

定数を除く:

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

= - \int 4^{u} u^{2} d u

部分積分を適用する

= - (\frac{2 ^{2 u - 1} u ^{2}}{ln ( 2 )} - \int \frac{4 ^{u} u}{ln ( 2 )} d u)

\int \frac{4 ^{u} u}{ln ( 2 )} d u = \frac{1}{ln ( 2 )} (\frac{2 ^{2 u - 1} u}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot 2^{2 u - 2})

= - (\frac{2 ^{2 u - 1} u ^{2}}{ln ( 2 )} - \frac{1}{ln ( 2 )} (\frac{2 ^{2 u - 1} u}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot 2^{2 u - 2}))

代用を戻す

u = - x

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

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

integral of (2/x+1)^2

\int (\frac{2}{x} + 1)^{2} d x

integral of a^{ny}

\int a^{n y} d y

integral of 2x^3+3x^2-10x

\int 2 x^{3} + 3 x^{2} - 10 x d x

integral of (x^2+x+8)/(x^2+3x-4)

\int \frac{x ^{2} + x + 8}{x ^{2} + 3 x - 4} d x

integral of (12x^8-x)/(x^3)

\int \frac{12 x ^{8} - x}{x ^{3}} d x