To evaluate the indefinite integral

$\int \frac{x^4}{\sqrt{1 - x^{10}}} \, dx$

Step-by-Step Solution:

1. Substitution: Let $u = x^5$. Then: $du = 5x^4 \, dx \quad \text{or} \quad \frac{1}{5} du = x^4 \, dx$ Also, notice that $x^{10} = (x^5)^2 = u^2$.

2. Rewriting the integral: Substituting $x^5 = u$ and $x^4 \, dx = \frac{1}{5} du$, the integral becomes: $\int \frac{x^4}{\sqrt{1 - x^{10}}} \, dx = \int \frac{\frac{1}{5} du}{\sqrt{1 - u^2}}$

3. Simplifying: Factor out the constant $\frac{1}{5}$: $\frac{1}{5} \int \frac{1}{\sqrt{1 - u^2}} \, du$

4. Recognizing the standard integral: The integral $\int \frac{1}{\sqrt{1 - u^2}} \, du$ is a standard form, which equals $\arcsin(u)$.

   Thus: $\frac{1}{5} \int \frac{1}{\sqrt{1 - u^2}} \, du = \frac{1}{5} \arcsin(u) + C$

5. Back-substitution: Recall that $u = x^5$, so: $\frac{1}{5} \arcsin(u) + C = \frac{1}{5} \arcsin(x^5) + C$

Final Answer:

$\int \frac{x^4}{\sqrt{1 - x^{10}}} \, dx = \frac{1}{5} \arcsin(x^5) + C$

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Would you like further clarification on any of the steps?

Related Questions:

1. How is the substitution method applied in integral problems?
2. What is the derivation of the formula $\int \frac{1}{\sqrt{1 - x^2}} \, dx = \arcsin(x) + C$?
3. How does substitution help simplify integrals with powers and roots?
4. Can you provide a geometrical interpretation of $\arcsin(x^5)$?
5. What are other techniques for solving integrals involving square roots?

Tip:

When encountering integrals with $\sqrt{1 - x^n}$, consider substitution to simplify powers or recognize standard trigonometric forms.