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人気のある微分積分 >

integral of 2^{-sqrt(x)}

解

\int 2^{- x} d x

解

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

解答ステップ

\int 2^{- x} d x

u置換積分法を適用する

= \int 2^{- u + 1} u d u

部分積分を適用する

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

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

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

代用を戻す

u = x

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

簡素化

\frac{( x ) ^{2}}{2 ^{x}} - - \frac{- ln ^{2} ( 2 ) ( x ) ^{2} + 2 ^{1 + x} ( - \frac{l n ( 2 ) x}{2 ^{x}} - \frac{1}{2 ^{x}} )}{ln ^{2} ( 2 ) \cdot 2 ^{x}} : \frac{x}{2 ^{x}} + \frac{- ln ^{2} ( 2 ) x + 2 ^{1 + x} ( - \frac{l n ( 2 ) x}{2 ^{x}} - \frac{1}{2 ^{x}} )}{ln ^{2} ( 2 ) \cdot 2 ^{x}}

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

定数を解答に追加する

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

グラフ

人気の例

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\int (t^{3} + t) d t

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\int ln (a + x) d x

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