https://file.aiquickdraw.com/mathbot/file/17342837443982zdq30i4.jpg

To solve the integral in the given image:

### Problem Statement:
We are solving the definite integral:
\[
\int_{\theta}^{x} \frac{\theta^\alpha}{\alpha \, y^{\alpha+1}} \, dy
\]

### Step-by-Step Solution:
1. **Extract constants**: Since \(\frac{\theta^\alpha}{\alpha}\) is independent of \(y\), it can be factored out:
   \[
   \frac{\theta^\alpha}{\alpha} \int_{\theta}^{x} \frac{1}{y^{\alpha+1}} \, dy
   \]

2. **Simplify the integral**:
   Rewrite \(\frac{1}{y^{\alpha+1}}\) as \(y^{-(\alpha+1)}\). Thus, the integral becomes:
   \[
   \frac{\theta^\alpha}{\alpha} \int_{\theta}^{x} y^{-(\alpha+1)} \, dy
   \]

3. **Integrate the power function**:
   The integral of \(y^{-(\alpha+1)}\) is given by:
   \[
   \int y^{-(\alpha+1)} \, dy = \frac{y^{-\alpha}}{-\alpha}
   \]
   So:
   \[
   \int_{\theta}^{x} y^{-(\alpha+1)} \, dy = \left[ \frac{y^{-\alpha}}{-\alpha} \right]_{\theta}^{x}
   \]

4. **Evaluate the definite integral**:
   Substitute the limits \(y = x\) and \(y = \theta\):
   \[
   \left[ \frac{y^{-\alpha}}{-\alpha} \right]_{\theta}^{x} = \frac{x^{-\alpha}}{-\alpha} - \frac{\theta^{-\alpha}}{-\alpha}
   \]

   Simplify the subtraction:
   \[
   = \frac{-x^{-\alpha} + \theta^{-\alpha}}{-\alpha} = \frac{\theta^{-\alpha} - x^{-\alpha}}{\alpha}
   \]

5. **Multiply by the constant**:
   Recall the constant factor \(\frac{\theta^\alpha}{\alpha}\):
   \[
   \frac{\theta^\alpha}{\alpha} \cdot \frac{\theta^{-\alpha} - x^{-\alpha}}{\alpha}
   \]

   Simplify:
   \[
   \frac{\theta^\alpha \cdot \theta^{-\alpha}}{\alpha^2} - \frac{\theta^\alpha \cdot x^{-\alpha}}{\alpha^2}
   \]

   Using \(\theta^\alpha \cdot \theta^{-\alpha} = 1\):
   \[
   \frac{1}{\alpha^2} - \frac{\theta^\alpha x^{-\alpha}}{\alpha^2}
   \]

6. **Final Answer**:
   \[
   \boxed{\frac{1}{\alpha^2} \left(1 - \theta^\alpha x^{-\alpha} \right)}
   \]

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Let me know if you'd like further clarifications or details!  

### Related Questions:
1. How does changing the limits of integration affect the solution?
2. What happens when \(\alpha = 1\)? How does the integral simplify?
3. Can this solution be generalized for a non-constant \(\theta\)?
4. What is the geometric interpretation of this integral?
5. How does this result behave as \(x \to \infty\)?

### Tip:
When solving integrals involving powers, always check if constants can be factored out to simplify calculations early.

Solve the given integral: ∫[θ to x] (θ^α / (α y^(α+1))) dy.

This solution provides a detailed step-by-step approach to solving the integral ∫[θ to x] (θ^α / (α y^(α+1))) dy. Using integration techniques and the power rule, the final result is simplified to a clear expression. Perfect for undergraduate calculus students.

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