What is the integral of 10^{-3x} ?

The integral of 10^{-3x} is -(1000^{-x})/(3ln(10))+C

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integral of 10^{-3x}

\int 1 0^{- 3 x} d x

- \frac{100 0 ^{- x}}{3 ln ( 10 )} + C

Solution steps

\int 1 0^{- 3 x} d x

Simplify

1 0^{- 3 x} : 100 0^{- x}

= \int 100 0^{- x} d x

Apply u-substitution

: \int - 100 0^{u} d u

= \int - 100 0^{u} d u

Take the constant out:

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

= - \int 100 0^{u} d u

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

= - \frac{100 0 ^{u}}{ln ( 1000 )}

Substitute back

u = - x

= - \frac{100 0 ^{- x}}{ln ( 1000 )}

Simplify

ln (1000) : 3 ln (10)

= - \frac{100 0 ^{- x}}{3 ln ( 10 )}

Add a constant to the solution

= - \frac{100 0 ^{- x}}{3 ln ( 10 )} + C

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What is the integral of 10^{-3x} ?
The integral of 10^{-3x} is -(1000^{-x})/(3ln(10))+C