The given integral involves the expression $x^2 - 25$, which suggests that we can use trigonometric substitution to simplify the integral. Specifically, because $x^2 - 25$ takes the form $a^2 - b^2$, we can use a substitution based on a trigonometric identity.

Step-by-step solution:

1. Recognize the substitution:

We have: $x^2 - 25 = (x - 5)(x + 5)$ This resembles the difference of squares. To handle this type of integral, we can use the substitution: $x = 5\sec(\theta)$ This works because: $x^2 - 25 = 25(\sec^2(\theta) - 1) = 25\tan^2(\theta)$ So, our substitution will transform $x^2 - 25$ into a form involving $\tan^2(\theta)$, which is easier to integrate.

2. Differentiate to find $dx$:

$dx = 5\sec(\theta)\tan(\theta) \, d\theta$

3. Substitute into the integral:

We substitute $x = 5\sec(\theta)$ and $dx = 5\sec(\theta)\tan(\theta) \, d\theta$ into the original integral $\int x^4 \, dx / (x^2 - 25)$.

$\int \frac{(5\sec(\theta))^4 \cdot 5\sec(\theta)\tan(\theta) \, d\theta}{25\tan^2(\theta)}$

4. Simplify:

$= \int \frac{625\sec^5(\theta)\tan(\theta) \cdot 5 \sec(\theta) \, d\theta}{25\tan^2(\theta)}$ $= \int \frac{625 \cdot 5 \sec^6(\theta)\tan(\theta) \, d\theta}{25\tan^2(\theta)}$ $= \int \frac{125 \sec^6(\theta)\, d\theta}{\tan(\theta)}$

At this point, the next steps would involve simplifying and performing the integration with respect to $\theta$, followed by reversing the substitution to return to $x$.

Do you want me to continue with the integration and details, or would you like to clarify anything?

Related questions:

1. How do we handle integrals involving different types of trigonometric substitutions?
2. Can you explain why we use $x = 5\sec(\theta)$ in this case and not another substitution?
3. How do we reverse the trigonometric substitution to get the final answer in terms of $x$?
4. What are other common trigonometric identities used in integral calculus?
5. How does the choice of trigonometric substitution simplify complicated integrals?

Tip:

Always check for patterns like $a^2 - x^2$ or $x^2 - a^2$, which suggest the use of trigonometric substitutions like sine or secant.