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

The integral of e^{-3x} is -1/3 e^{-3x}+C

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integral of e^{-3x}

\int e^{- 3 x} d x

- \frac{1}{3} e^{- 3 x} + C

Solution steps

\int e^{- 3 x} d x

Apply u-substitution

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

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

Use the common integral:

\int e^{u} d u = e^{u}

= - \frac{1}{3} e^{u}

Substitute back

u = - 3 x

= - \frac{1}{3} e^{- 3 x}

Add a constant to the solution

= - \frac{1}{3} e^{- 3 x} + C

\int \frac{41}{x ^{3} - 27} d x

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\frac{\partial}{\partial y} (3 cos (x))

x \to 3 lim (\frac{( x - 3 )}{x - 3})