What is the integral of x(2^{-x}) ?

The integral of x(2^{-x}) is -(2^{-x}x)/(ln(2))-1/(ln^2(2))*2^{-x}+C

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integral of x(2^{-x})

\int x (2^{- x}) d x

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

Solution steps

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

Apply Integration By Parts

: - \frac{2 ^{- x} x}{ln ( 2 )} - \int - \frac{2 ^{- x}}{ln ( 2 )} d x

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

\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}

Add a constant to the solution

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

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What is the integral of x(2^{-x}) ?
The integral of x(2^{-x}) is -(2^{-x}x)/(ln(2))-1/(ln^2(2))*2^{-x}+C