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

Let's integrate:

\int e^{3 x} d x

For the math expression:

e^{3 x}

, knowing that (exp(a))' = a'*exp(a), let's try differentiating

\exp (3 x)

Let's differentiate:

\frac{d}{d x} (\exp (3 x))

Apply the differentiation rule:

\frac{d}{d x} (\exp (f)) = \frac{d}{d x} (f) \cdot \exp (f)

with f =

3 x

ⓘ Differentiating

\exp (x)

gives

\exp (x)

\frac{d}{d x} (\exp (3 x))

\frac{d}{d x} (3 x) \cdot \exp (3 x)

\frac{d}{d x} (3 x) \cdot \exp (3 x)

ⓘ A constant factor stays in front of the derivative: that of

k u

k \cdot u'

\frac{d}{d x} (3 x)

3

3 \exp (3 x)

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

ⓘ Differentiating

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

gives

e^{3 x}

, so

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

is an antiderivative of

e^{3 x}

\int e^{3 x} d x

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

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

ⓘ Multiplying a fraction by a term multiplies its numerator:

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

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

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

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

▶An indefinite integral is defined up to an additive constant C; we therefore obtain:

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