What is the integral of 5^{-4(x-7)} ?

The integral of 5^{-4(x-7)} is -(625^{-x+7})/(4ln(5))+C

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integral of 5^{-4(x-7)}

\int 5^{- 4 (x - 7)} d x

- \frac{62 5 ^{- x + 7}}{4 ln ( 5 )} + C

Solution steps

\int 5^{- 4 (x - 7)} d x

Simplify

5^{- 4 (x - 7)} : 62 5^{- x + 7}

= \int 62 5^{- x + 7} d x

Apply u-substitution

: \int - 62 5^{u} d u

= \int - 62 5^{u} d u

Take the constant out:

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

= - \int 62 5^{u} d u

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

= - \frac{62 5 ^{u}}{ln ( 625 )}

Substitute back

u = - x + 7

= - \frac{62 5 ^{- x + 7}}{ln ( 625 )}

Simplify

ln (625) : 4 ln (5)

= - \frac{62 5 ^{- x + 7}}{4 ln ( 5 )}

Add a constant to the solution

= - \frac{62 5 ^{- x + 7}}{4 ln ( 5 )} + C

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What is the integral of 5^{-4(x-7)} ?
The integral of 5^{-4(x-7)} is -(625^{-x+7})/(4ln(5))+C