What is the integral of-2^{-5x} ?

The integral of-2^{-5x} is (32^{-x})/(5ln(2))+C

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integral of-2^{-5x}

\int - 2^{- 5 x} d x

\frac{3 2 ^{- x}}{5 ln ( 2 )} + C

Solution steps

\int - 2^{- 5 x} d x

Take the constant out:

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

= - \int 2^{- 5 x} d x

Apply u-substitution

: \int - \frac{1}{5} \cdot 2^{u} d u

= - \int - \frac{1}{5} \cdot 2^{u} d u

Take the constant out:

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

= - (- \frac{1}{5} \cdot \int 2^{u} d u)

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

= - (- \frac{1}{5} \cdot \frac{2 ^{u}}{ln ( 2 )})

Substitute back

u = - 5 x

= - (- \frac{1}{5} \cdot \frac{2 ^{- 5 x}}{ln ( 2 )})

Simplify

- (- \frac{1}{5} \cdot \frac{2 ^{- 5 x}}{ln ( 2 )}) : \frac{3 2 ^{- x}}{5 ln ( 2 )}

= \frac{3 2 ^{- x}}{5 ln ( 2 )}

Add a constant to the solution

= \frac{3 2 ^{- x}}{5 ln ( 2 )} + C

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What is the integral of-2^{-5x} ?
The integral of-2^{-5x} is (32^{-x})/(5ln(2))+C