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To solve the integral \(\int \frac{\sin\left(\sqrt[3]{x}\right)}{\sqrt[3]{x^2}} \, dx\) using substitution, follow these steps:

### Step 1: Choose a Substitution
Let:
\[
u = \sqrt[3]{x}
\]
Then:
\[
u^3 = x
\]
Differentiate both sides with respect to \(x\):
\[
3u^2 \, du = dx
\]

### Step 2: Rewrite the Integral
Substitute the values into the integral:
\[
\int \frac{\sin(u)}{u^2} \cdot 3u^2 \, du
\]

Simplify the integrand:
\[
3 \int \sin(u) \, du
\]

### Step 3: Integrate
The integral of \(\sin(u)\) with respect to \(u\) is:
\[
-\cos(u)
\]
Thus, the integral becomes:
\[
-3 \cos(u) + C
\]

### Step 4: Substitute Back \(u = \sqrt[3]{x}\)
Finally, replace \(u\) with \(\sqrt[3]{x}\) to obtain the solution:
\[
\int \frac{\sin\left(\sqrt[3]{x}\right)}{\sqrt[3]{x^2}} \, dx = -3 \cos\left(\sqrt[3]{x}\right) + C
\]

This is the final answer.

Would you like further details or have any other questions?

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Here are five related questions to explore further:
1. How would the integral change if \(\sin\left(\sqrt[3]{x}\right)\) were replaced with \(\cos\left(\sqrt[3]{x}\right)\)?
2. What if the exponent in the substitution was different, for example, \(u = \sqrt[5]{x}\)?
3. How does the process differ for a definite integral with the same integrand?
4. Can you use the same substitution method to solve an integral involving \(\tan\) or \(\sec\) instead of \(\sin\)?
5. What is the effect of changing the power of \(x\) in the denominator?

**Tip:** When using substitution in integrals, always ensure the derivative of the substitution aligns with the original differential \(dx\) to simplify the integrand properly.

Learn how to solve the integral ∫ sin(sqrt[3]{x}) / sqrt[3]{x^2} using substitution method. This detailed step-by-step solution covers integration techniques, including substitution and trigonometric integrals. Suitable for advanced high school or college students.

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