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

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

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integral of a^{-3x}

\int a^{- 3 x} d x

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

Solution steps

\int a^{- 3 x} d x

Apply u-substitution

: \int - \frac{1}{3} a^{u} d u

= \int - \frac{1}{3} a^{u} d u

Take the constant out:

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

= - \frac{1}{3} \cdot \int a^{u} d u

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

= - \frac{1}{3} \cdot \frac{a ^{u}}{ln ( a )}

Substitute back

u = - 3 x

= - \frac{1}{3} \cdot \frac{a ^{- 3 x}}{ln ( a )}

Simplify

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

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

Add a constant to the solution

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

\int \frac{e ^{3 x}}{( 4 + e ^{2 x} ) ^{2}} d x

f (x) = \frac{8}{x + 11}, at x = 5

\int_{0}^{1} 3 x d x

x y^{^{'}} = x - y

(x - 1)^{2}, x + 1

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