https://file.aiquickdraw.com/mathbot/file/1725637169657781x5k8g.jpg

The integral in the image is:

\[
\int \frac{x^4}{x^{10} + 1} \, dx
\]

To solve this, we can try the substitution method.

Let:
\[
u = x^{10} + 1
\]
Then:
\[
du = 10x^9 \, dx
\]

Now, we need to express the original integrand in terms of \( u \). However, the substitution doesn't immediately simplify the numerator \( x^4 \), so we need to consider breaking the expression down.

Let's modify the approach and rewrite the integrand. I'll work through the next steps systematically.The indefinite integral evaluates to:

\[
\int \frac{x^4}{x^{10} + 1} \, dx = \frac{1}{5} \arctan(x^5)
\]

This result uses the arctangent function due to the structure of the denominator.

Would you like more details on any step or have any questions?

Here are 5 related questions to expand your understanding:
1. How does substitution simplify integrals involving polynomials?
2. What is the role of the arctangent function in integration?
3. How would the result differ if the power in the denominator were changed, say \( x^{12} + 1 \)?
4. Can this type of integral be evaluated using partial fractions?
5. How does this integration compare to solving integrals with exponential functions?

**Tip**: Whenever you encounter a rational function like this, consider using substitution and recognize common forms like \( \arctan \) to simplify the problem.

Evaluate the indefinite integral: ∫(x^4 / (x^10 + 1)) dx.

Find the indefinite integral of ∫(x^4 / (x^10 + 1)) dx using substitution and recognize the role of the arctangent function in the solution. This calculus problem illustrates the application of trigonometric integrals and inverse trigonometric functions. Suitable for high school seniors or college students.

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