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

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

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integral of-2^{-x}

\int - 2^{- x} d x

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

Solution steps

\int - 2^{- x} d x

Take the constant out:

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

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

Apply u-substitution

: \int - 2^{u} d u

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

Take the constant out:

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

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

\int a^{x} d x = \frac{a ^{x}}{ln a}

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

Substitute back

u = - x

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

Simplify

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

Add a constant to the solution

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

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