what if instead i sub u = x^3/2 and then take u = a*sin(deta)

Using the substitution \( u = x^{3/2} \) and then substituting \( u = a \sin \theta \) can be a more efficient way to solve this integral. Let's follow this method step-by-step.

### Step-by-Step Solution:

1. **Substitution \( u = x^{3/2} \)**:
   \[
   u = x^{3/2} \implies du = \frac{3}{2} x^{1/2} \, dx \implies dx = \frac{2}{3} x^{-1/2} \, du
   \]
   Since \( x = (u)^{2/3} \),
   \[
   dx = \frac{2}{3} (u^{2/3})^{-1/2} \, du = \frac{2}{3} u^{-1/3} \, du
   \]

2. **Rewrite the integral**:
   \[
   \int \frac{x^{3/2}}{\sqrt{a^2 - x}} \, dx = \int \frac{u}{\sqrt{a^2 - (u)^{2/3}}} \cdot \frac{2}{3} u^{-1/3} \, du = \frac{2}{3} \int \frac{u^{2/3}}{\sqrt{a^2 - u^{2/3}}} \, du
   \]

3. **Simplify the integral**:
   \[
   \frac{2}{3} \int \frac{u^{2/3}}{\sqrt{a^2 - u^{2/3}}} \, du
   \]

4. **Second substitution \( u = a \sin \theta \)**:
   \[
   u = a \sin \theta \implies du = a \cos \theta \, d\theta
   \]
   The integral becomes:
   \[
   \frac{2}{3} \int \frac{(a \sin \theta)^{2/3}}{\sqrt{a^2 - (

Learn how to solve integrals using the substitution method with u = x^(3/2) and further u = a*sin(theta). This detailed solution covers advanced integration techniques and trigonometric substitutions, suitable for college-level mathematics.

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