integral from 0 to 1 of x2^{-x}

Lösung

\int_{0}^{1} x 2^{- x} d x

\frac{- ln ( 2 ) + 1}{2 ln ^{2} ( 2 )}

Dezimale

0.31933 \dots

Schritte zur Lösung

\int_{0}^{1} x \cdot 2^{- x} d x

Wende die partielle Integration an

= [- \frac{2 ^{- x} x}{ln ( 2 )} - \int - \frac{2 ^{- x}}{ln ( 2 )} d x]_{0}^{1}

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

= [- \frac{2 ^{- x} x}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot 2^{- x}]_{0}^{1}

Berechne die Grenzen:

\frac{- ln ( 2 ) + 1}{2 ln ^{2} ( 2 )}

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

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