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

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

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integral of 5^{-x}

\int 5^{- x} d x

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

Solution steps

\int 5^{- x} d x

Apply u-substitution

: \int - 5^{u} d u

= \int - 5^{u} d u

Take the constant out:

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

= - \int 5^{u} d u

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

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

Substitute back

u = - x

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (\frac{y}{x ^{2} ( y + 3 )})

\frac{d y}{d x} = (x + y - 1)^{3} - 1

\int (sin (x) + 3) d x

\int e^{6}

y = cos (4 x)