To solve the integral \(\int x^5 \cdot \frac{1}{\sqrt{x^4 + 4}} \, dx\), we can proceed with a substitution method.

### Step 1: Substitution
Let \( u = x^4 + 4 \). Then, we have:
\[
\frac{du}{dx} = 4x^3
\]
Thus, \( du = 4x^3 \, dx \) or \( \frac{du}{4x^3} = dx \).

### Step 2: Rewriting the Integral
We substitute \( u = x^4 + 4 \) into the integral. The integral becomes:
\[
\int x^5 \cdot \frac{1}{\sqrt{u}} \cdot \frac{du}{4x^3}
\]

Simplify the expression:
\[
\frac{1}{4} \int x^2 \cdot \frac{1}{\sqrt{u}} \, du
\]

### Step 3: Express \( x^2 \) in terms of \( u \)
Since \( u = x^4 + 4 \), we have \( x^4 = u - 4 \). Therefore, \( x^2 = \sqrt{u - 4} \).

Substitute back into the integral:
\[
\frac{1}{4} \int \sqrt{u - 4} \cdot \frac{1}{\sqrt{u}} \, du
\]

This simplifies to:
\[
\frac{1}{4} \int \sqrt{\frac{u - 4}{u}} \, du
\]

Further simplifying:
\[
\frac{1}{4} \int \sqrt{1 - \frac{4}{u}} \, du
\]

### Step 4: Trigonometric or Further Substitution (if possible)
At this point, the integral becomes quite complex, and standard trigonometric substitution methods may be necessary, but it appears to be a non-standard integral that might require special functions or numerical methods for a complete solution.

Would you like to explore the next steps, or would you prefer an approximation or numerical evaluation?

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**Relative Questions:**
1. How do we handle integrals involving trigonometric substitutions?
2. What are some common techniques for solving integrals with square roots?
3. Can all integrals be solved analytically?
4. How does substitution simplify the process of integration?
5. What are special functions, and when are they used in integration?

**Tip:** Always check if a substitution can simplify the integrand before attempting more complex methods.

Learn how to integrate x^5 * 1/(sqrt(x^4 + 4)) dx with a detailed step-by-step solution using substitution and trigonometric methods. Understand the application of special functions in non-standard integrals. Suitable for advanced students and professionals.

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