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

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

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integral of-e^{3t}

\int - e^{3 t} d t

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

Solution steps

\int - e^{3 t} d t

Take the constant out:

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

= - \int e^{3 t} d t

Apply u-substitution

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

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

Add a constant to the solution

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