Let's solve the integral:

$\int_1^3 \frac{4}{x^4} \, dx$

Step 1: Simplify the integral

We can rewrite the integrand as:

$\frac{4}{x^4} = 4x^{-4}$

Now the integral becomes:

$\int_1^3 4x^{-4} \, dx$

Step 2: Apply the power rule for integration

The power rule for integration is:

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(where n \neq -1)}

For $x^{-4}$, the integral is:

$\int x^{-4} \, dx = \frac{x^{-3}}{-3} = -\frac{1}{3x^3}$

Multiplying by 4, the integral becomes:

$\int 4x^{-4} \, dx = -\frac{4}{3x^3}$

Step 3: Evaluate the definite integral

We now evaluate the integral from 1 to 3:

$\left[ -\frac{4}{3x^3} \right]_1^3$

First, substitute the upper limit (3):

$-\frac{4}{3(3^3)} = -\frac{4}{81}$

Now, substitute the lower limit (1):

$-\frac{4}{3(1^3)} = -\frac{4}{3}$

Step 4: Subtract the values

Now, subtract the value at $x = 3$ from the value at $x = 1$:

$-\frac{4}{81} - \left(-\frac{4}{3}\right) = -\frac{4}{81} + \frac{4}{3}$

Step 5: Simplify the result

To combine the fractions, we need a common denominator. The least common denominator between 81 and 3 is 81, so:

$\frac{4}{3} = \frac{4 \times 27}{3 \times 27} = \frac{108}{81}$

Now, combine the fractions:

$-\frac{4}{81} + \frac{108}{81} = \frac{-4 + 108}{81} = \frac{104}{81}$

Final Answer:

The value of the definite integral is:

$\int_1^3 \frac{4}{x^4} \, dx = \frac{104}{81}$

Would you like further clarification on any step or more details?

Here are five related questions you might find interesting:

1. How would the integral change if the power of $x$ in the denominator were different?
2. Can you apply this method to an integral of the form $\int_1^3 \frac{k}{x^n} \, dx$ for any constant $k$?
3. How does the Fundamental Theorem of Calculus help with solving definite integrals?
4. What would the result be if the limits of integration were reversed (e.g., from 3 to 1)?
5. Can you apply the same approach to integrals involving other powers or functions of $x$?

Tip: Always simplify the integrand as much as possible before applying the rules of integration.