integrate e^(3x) — step-by-step solution

Let's integrate:

\int(e^{(3 x)})dx

We attempt to reverse an integration: starting from

e^{(3 x)}

and

{(\exp (a))}^{'} = \exp (a)

, we differentiate using exp

Let's perform the differentiation:

{(\exp (3 x))}^{'}

Apply the differentiation rule:

{(\exp (f))}^{'} = {(f)}^{'} \cdot \exp (f)

with f =

3 x

{(3 x)}^{'} \cdot \exp (3 x)

3 \exp (3 x)

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

\frac{1}{3} \cdot e^{(3 x)}

\frac{e^{(3 x)}}{3}

Indefinite integrals are defined up to an additive constant C, so this yields:

\frac{e^{(3 x)}}{3} + C

▷Result: $\frac{e^{(3 x)}}{3} + C$ , where C is a constant