What is the integral of 3^{-2x} ?

The integral of 3^{-2x} is -(9^{-x})/(2ln(3))+C

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integral of 3^{-2x}

\int 3^{- 2 x} d x

- \frac{9 ^{- x}}{2 ln ( 3 )} + C

Solution steps

\int 3^{- 2 x} d x

Apply u-substitution

: \int - \frac{1}{2} \cdot 3^{u} d u

= \int - \frac{1}{2} \cdot 3^{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 3^{u} d u

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

= - \frac{1}{2} \cdot \frac{3 ^{u}}{ln ( 3 )}

Substitute back

u = - 2 x

= - \frac{1}{2} \cdot \frac{3 ^{- 2 x}}{ln ( 3 )}

Simplify

- \frac{1}{2} \cdot \frac{3 ^{- 2 x}}{ln ( 3 )} : - \frac{9 ^{- x}}{2 ln ( 3 )}

= - \frac{9 ^{- x}}{2 ln ( 3 )}

Add a constant to the solution

= - \frac{9 ^{- x}}{2 ln ( 3 )} + C

\int \frac{2 x ^{2} - 2 x - 8}{x ^{2} ( x - 4 )} d x

x \to 0 - lim (cos (\frac{1}{x}))

\int \frac{1}{1 ^{2} + 1 x ^{2}} d x

\frac{d}{d x} (\frac{x ^{2} + 100}{x})

x \to 2 lim (ln (2))

What is the integral of 3^{-2x} ?
The integral of 3^{-2x} is -(9^{-x})/(2ln(3))+C