# How do you evaluate the integral inte^(4x) dx?

Aug 13, 2014

We will use $u$-substitution, letting $u = 4 x$.

Thus, $\mathrm{du} = 4 \mathrm{dx}$.

Also, we will use the constant law of integration, namely $\int C \cdot f \left(x\right) \mathrm{dx} = C \cdot \int f \left(x\right) \mathrm{dx}$ to rewrite the integral so that it contains $\mathrm{du}$:

$\int {e}^{4 x} \mathrm{dx} = \frac{1}{4} \int 4 \cdot {e}^{4 x} \mathrm{dx}$

Now, we will rewrite in terms of $u$:

$\int {e}^{4 x} \mathrm{dx} = \frac{1}{4} \int {e}^{u} \mathrm{du}$

We know that the integral of ${e}^{u} \mathrm{du}$ will simply be ${e}^{u}$. Remember the constant of integration:

$\int {e}^{4 x} \mathrm{dx} = \frac{1}{4} {e}^{u} + C$

Substituting back $u$ gives:

$\int {e}^{4 x} \mathrm{dx} = \frac{1}{4} {e}^{4 x} + C$