To solve the integral \(\int_2^4 \frac{\sqrt{x^2 - 4}}{x^4} \, dx\), we will follow these steps:
### Step 1: Simplify the integrand
We start with the integral:
\[
\int_2^4 \frac{\sqrt{x^2 - 4}}{x^4} \, dx
\]
We can rewrite the integrand by factoring out \(x^2\) from the square root:
\[
\sqrt{x^2 - 4} = \sqrt{x^2(1 - \frac{4}{x^2})} = x \sqrt{1 - \frac{4}{x^2}}
\]
Thus, the integral becomes:
\[
\int_2^4 \frac{x \sqrt{1 - \frac{4}{x^2}}}{x^4} \, dx = \int_2^4 \frac{\sqrt{1 - \frac{4}{x^2}}}{x^3} \, dx
\]
### Step 2: Substitution
Next, we will use the substitution \(t = 1 - \frac{4}{x^2}\). To find \(dt\), we differentiate:
\[
dt = \frac{d}{dx}(1 - \frac{4}{x^2}) = \frac{8}{x^3} \, dx \implies dx = \frac{x^3}{8} \, dt
\]
Now, we need to express \(x\) in terms of \(t\):
\[
t = 1 - \frac{4}{x^2} \implies \frac{4}{x^2} = 1 - t \implies x^2 = \frac{4}{1 - t} \implies x = \frac{2}{\sqrt{1 - t}}
\]
### Step 3: Change the limits of integration
When \(x = 2\):
\[
t = 1 - \frac{4}{2^2} = 1 - 1 = 0
\]
When \(x = 4\):
\[
t = 1 - \frac{4}{4^2} = 1 - \frac{4}{16} = 1 - \frac{1}{4} = \frac{3}{4}
\]
Thus, the limits change from \(x = 2\) to \(x = 4\) into \(t = 0\) to \(t = \frac{3}{4}\).
### Step 4: Substitute into the integral
Now substituting everything back into the integral:
\[
\int_0^{\frac{3}{4}} \sqrt{t} \cdot \frac{2}{\sqrt{1 - t}} \cdot \frac{2}{8} \, dt = \frac{1}{2} \int_0^{\frac{3}{4}} \frac{2\sqrt{t}}{\sqrt{1 - t}} \, dt
\]
### Step 5: Solve the integral
The integral \(\int \frac{\sqrt{t}}{\sqrt{1 - t}} \, dt\) can be solved using the beta function or trigonometric substitution. However, we can also use the formula for the integral of the form:
\[
\int_0^a t^{m-1} (1-t)^{n-1} \, dt = \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)}
\]
For our case, we have \(m = \frac{3}{2}\) and \(n = \frac{1}{2}\):
\[
\int_0^{\frac{3}{4}} t^{\frac{1}{2}} (1 - t)^{-\frac{1}{2}} \, dt = B\left(\frac{3}{4}, \frac{1}{2}\right)
\]
### Step 6: Evaluate the final result
Evaluating this integral gives us:
\[
\frac{1}{2} \cdot \frac{2\sqrt{3}}{3} = \frac{\sqrt{3}}{3}
\]
Finally, we multiply by the constant factor we factored out earlier, leading to:
\[
\text{Final result} = \frac{32\sqrt{3}}{3}
\]
