Thank you for the correction. The integral to evaluate is:

\[
\int_{0}^{\infty} x \sin\left((bx)^{\frac{1}{n}}\right) \, dx
\]

Let's solve this integral step by step.

### Step-by-Step Solution:

1. **Substitution**:
   Let \( u = (bx)^{\frac{1}{n}} \). Then:
   \[
   x = \left(\frac{u}{b}\right)^n
   \]
   and the differential:
   \[
   du = \frac{1}{n}(bx)^{\frac{1}{n} - 1} b \, dx = \frac{b}{n} \left(\frac{u}{b}\right)^{1-\frac{1}{n}} dx
   \]
   Solving for \( dx \):
   \[
   dx = \frac{n}{b} u^{n-1} \, du
   \]
   
   The limits of integration change accordingly:
   - When \( x = 0 \), \( u = 0 \)
   - When \( x \to \infty \), \( u \to \infty \)

   Substituting into the integral:
   \[
   \int_{0}^{\infty} x \sin\left((bx)^{\frac{1}{n}}\right) \, dx = \int_{0}^{\infty} \left(\frac{u}{b}\right)^n \sin(u) \frac{n}{b} u^{n-1} \, du
   \]
   
   Simplifying:
   \[
   \frac{n}{b^{n+1}} \int_{0}^{\infty} u^{2n-1} \sin(u) \, du
   \]

2. **Standard Integral**:
   The integral \( \int_{0}^{\infty} u^{2n-1} \sin(u) \, du \) is a well-known standard integral that can be solved using the Gamma function and sine integral properties:

   \[
   \int_{0}^{\infty} u^{2n-1} \sin(u) \, du = (-1)^{n-1} \frac{(2n-1)!}{2}
   \]
   
   So the integral simplifies to:
   \[
   \frac{n}{b^{n+1}} \cdot (-1)^{n-1} \cdot \frac{(2n-1)!}{2}
   \]

3. **Final Answer**:
   Combining the above results, the integral evaluates to:
   \[
   \int_{0}^{\infty} x \sin\left((bx)^{\frac{1}{n}}\right) \, dx = \frac{(-1)^{n-1} n (2n-1)!}{2 b^{n+1}}
   \]

### Conclusion:
The integral evaluates to \( \frac{(-1)^{n-1} n (2n-1)!}{2 b^{n+1}} \).

Would you like further details or have any questions about the solution? Here are some related questions to consider:

1. How do factorials and Gamma functions relate to each other?
2. What other standard integrals involve trigonometric functions and powers of variables?
3. Why is substitution a useful technique in integration?
4. What are the conditions under which specific standard integrals can be applied?
5. How do different substitutions affect the limits of integration?

**Tip:** Always verify the structure of the function inside an integral before proceeding with integration techniques like substitution or integration by parts.

Learn how to evaluate the integral of x sin((bx)^(1/n)) from 0 to Infinity with a detailed step-by-step solution. This problem involves substitution and the integral of sine function. Suitable for advanced undergraduate mathematics students.

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