The given integral is:
$\int_0^1 e^{-4x} \, dx.$

Step 1: Integrate $e^{-4x}$

The integral of $e^{ax}$ with respect to $x$ is:
\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C, \quad \text{where a is a constant}.

Here, $a = -4$. Thus:
$\int e^{-4x} \, dx = \frac{1}{-4} e^{-4x} = -\frac{1}{4} e^{-4x}.$

Step 2: Apply the limits $0$ to $1$

Using the Fundamental Theorem of Calculus:
$\int_0^1 e^{-4x} \, dx = \left[ -\frac{1}{4} e^{-4x} \right]_0^1.$

Substitute the upper and lower limits:
$= -\frac{1}{4} e^{-4(1)} - \left( -\frac{1}{4} e^{-4(0)} \right).$

Simplify each term:
$= -\frac{1}{4} e^{-4} + \frac{1}{4} e^{0}.$

Since $e^{0} = 1$:
$= -\frac{1}{4} e^{-4} + \frac{1}{4}.$

Combine terms:
$= \frac{1}{4} \left( 1 - e^{-4} \right).$

Final Answer:

$\int_0^1 e^{-4x} \, dx = \frac{1}{4} \left( 1 - e^{-4} \right).$

Let me know if you'd like further clarification!

Related Questions

1. How would the solution change if the limits were from $0$ to $2$?
2. What is the general formula for $\int e^{ax} \, dx$?
3. How can you approximate $e^{-4}$ using a calculator?
4. What happens to the integral if $-4x$ is replaced by $-kx$?
5. How would you solve this problem using numerical integration techniques?

Tip:

When dealing with exponential functions in integrals, remember to factor out constants for simplification.