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

The integral of 5^{-2x} is -(25^{-x})/(2ln(5))+C

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integral of 5^{-2x}

\int 5^{- 2 x} d x

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

Solution steps

\int 5^{- 2 x} d x

Apply u-substitution

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

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

Take the constant out:

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

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

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

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

Substitute back

u = - 2 x

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of 5^{-2x} ?
The integral of 5^{-2x} is -(25^{-x})/(2ln(5))+C