Integrate sin^-1(√1-t)/(√t) without trigonometric substitution 

To integrate the function \( \int \frac{\sin^{-1}(\sqrt{1-t})}{\sqrt{t}} \, dt \), we can use integration by parts. Let’s proceed with the integration step-by-step.

1. **Choose \( u \) and \( dv \)**:
   - Let \( u = \sin^{-1}(\sqrt{1-t}) \), hence \( du = \frac{1}{\sqrt{1-t}} \cdot \frac{1}{2\sqrt{1-t}}(-1) \, dt = -\frac{1}{2\sqrt{1-t}} \, dt \).
   - Let \( dv = \frac{1}{\sqrt{t}} \, dt \), hence \( v = 2\sqrt{t} \).

2. **Integration by Parts Formula**:
   The formula is given by:
   \[
   \int u \, dv = uv - \int v \, du
   \]

3. **Applying Integration by Parts**:
   \[
   \int \frac{\sin^{-1}(\sqrt{1-t})}{\sqrt{t}} \, dt = uv - \int v \, du
   \]
   Substituting \( u \) and \( v \):
   \[
   = 2\sqrt{t} \sin^{-1}(\sqrt{1-t}) - \int 2\sqrt{t} \left(-\frac{1}{2\sqrt{1-t}}\right) \, dt
   \]
   Simplifying:
   \[
   = 2\sqrt{t} \sin^{-1}(\sqrt{1-t}) + \int \frac{\sqrt{t}}{\sqrt{1-t}} \, dt
   \]

4. **Simplifying the Integral**:
   The integral \( \int \frac{\sqrt{t}}{\sqrt{1-t}} \, dt \) can be rewritten as:
   \[
   = \int \sqrt{\frac{t}{1-t}} \, dt
   \]
   Let \( t = \sin^2(\theta) \), then \( dt = 2\sin(\theta)\cos(\theta) \, d\theta \), and \( \sqrt{1-t} = \cos(\theta) \). This transforms the integral into:
   \[
   \int \sin(\theta) \, d\theta = -\cos(\theta) + C
   \]

5. **Back Substitute**:
   From \( t = \sin^2(\theta) \), we have \( \theta = \sin^{-1}(\sqrt{t}) \), so:
   \[
   \int \frac{\sqrt{t}}{\sqrt{1-t}} \, dt = -\sqrt{1-t} + C
   \]

6. **Final Expression**:
   Combining everything gives us:
   \[
   \int \frac{\sin^{-1}(\sqrt{1-t})}{\sqrt{t}} \, dt = 2\sqrt{t} \sin^{-1}(\sqrt{1-t}) - \sqrt{1-t} + C
   \]

Thus, the integral evaluates to:
\[
\int \frac{\sin^{-1}(\sqrt{1-t})}{\sqrt{t}} \, dt = 2\sqrt{t} \sin^{-1}(\sqrt{1-t}) - \sqrt{1-t} + C
\]

If you have any questions or need further details, let me know! Here are some related questions to consider:

1. What is the geometric interpretation of the inverse sine function?
2. How can integration by parts be applied to different types of integrals?
3. What are common integrals involving the square root functions?
4. Can you show examples of integrals using trigonometric substitution?
5. How do you determine when to use integration by parts versus substitution? 

**Tip**: Always carefully choose \( u \) and \( dv \) based on their derivatives to simplify the integral effectively.

Integrate sin^-1(√1-t)/(√t) without trigonometric substitution

Explore the integration of the function sin^-1(√1-t)/(√t) without trigonometric substitution. This detailed solution utilizes integration by parts and provides step-by-step guidance, ideal for students in Grades 11-12.

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