https://file.aiquickdraw.com/mathbot/file/1727310409371rgz81r8z.jpg

The integral in the image is:

\[
\int \frac{1}{\sqrt{x} \sqrt{1 - x}} \, dx
\]

This type of integral can be simplified using a standard trigonometric substitution. The appropriate substitution is:

\[
x = \sin^2(\theta)
\]

Let's break it down:

1. If \( x = \sin^2(\theta) \), then \( dx = 2\sin(\theta)\cos(\theta) \, d\theta \).
2. Additionally, \( \sqrt{x} = \sin(\theta) \) and \( \sqrt{1 - x} = \cos(\theta) \).

Using these substitutions, the integral simplifies into a standard form. Let me compute the exact result.

\[
\int \frac{1}{\sin(\theta) \cos(\theta)} \cdot 2 \sin(\theta)\cos(\theta) \, d\theta
\]

This simplifies to:

\[
\int 2 \, d\theta = 2\theta + C
\]

Since \( x = \sin^2(\theta) \), we can find \( \theta \) as \( \theta = \sin^{-1}(\sqrt{x}) \). Therefore, the final result is:

\[
2 \sin^{-1}(\sqrt{x}) + C
\]

Thus, the solution to the integral is:

\[
2 \sin^{-1}(\sqrt{x}) + C
\]

Would you like further details or have any questions about this solution?

Here are 5 related questions for further understanding:
1. How do trigonometric substitutions help solve integrals with square roots?
2. What other trigonometric substitutions could you use for different integrals?
3. Can this integral be solved using a different substitution method?
4. Why is the inverse sine function used in this solution?
5. How does the constant of integration affect the final answer?

**Tip:** Always check for symmetry in integrals involving square roots and trigonometric functions—this can simplify the problem significantly.

Learn how to solve the integral ∫ 1 / (sqrt(x) * sqrt(1 - x)) dx using trigonometric substitution. This problem involves key calculus concepts such as trigonometric substitutions, inverse trigonometric functions, and integration techniques. Suitable for college-level calculus students.

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