integral of 2^{-sqrt(x)}

Lösung

\int 2^{- x} d 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}} + C

Schritte zur Lösung

\int 2^{- x} d x

Wende U-Substitution an

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

Wende die partielle Integration an

= \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}}

Setze in

u = x

ein

= \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}}

Vereinfache

\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}}

Füge eine Konstante zur Lösung hinzu

= \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

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