We are asked to evaluate the integral:

$I = \int_0^1 \frac{1}{\sqrt{x(1 - x)}} \, dx$

Step 1: Simplification using a substitution

Let's simplify this integral by performing a common trigonometric substitution:

Let $x = \sin^2 \theta$, so that $dx = 2 \sin \theta \cos \theta \, d\theta$. Notice that:

• When $x = 0$, $\theta = 0$,
• When $x = 1$, $\theta = \frac{\pi}{2}$.

Now, we rewrite the integral in terms of $\theta$:

$\sqrt{x(1 - x)} = \sqrt{\sin^2 \theta (1 - \sin^2 \theta)} = \sqrt{\sin^2 \theta \cos^2 \theta} = \sin \theta \cos \theta$

Thus, the integral becomes:

$I = \int_0^{\frac{\pi}{2}} \frac{1}{\sin \theta \cos \theta} \cdot 2 \sin \theta \cos \theta \, d\theta = 2 \int_0^{\frac{\pi}{2}} d\theta = \pi$

Step 2: Test for Convergence

We want to check the convergence at the boundaries $x = 0$ and $x = 1$.

Behavior near $x = 0$:

As $x \to 0$, the integrand behaves like:

$\frac{1}{\sqrt{x}} \text{ since } (1 - x) \to 1$

The integral of $\frac{1}{\sqrt{x}}$ near 0 is:

$\int_0^\epsilon \frac{1}{\sqrt{x}} \, dx = 2 \sqrt{\epsilon}$

which converges as $\epsilon \to 0$.

Behavior near $x = 1$:

As $x \to 1$, the integrand behaves like:

$\frac{1}{\sqrt{1 - x}} \text{ since } x \to 1$

The integral of $\frac{1}{\sqrt{1 - x}}$ near 1 is:

$\int_{1 - \epsilon}^1 \frac{1}{\sqrt{1 - x}} \, dx = 2 \sqrt{\epsilon}$

which also converges as $\epsilon \to 0$.

Conclusion:

Both boundaries behave well, and the integral converges. Thus, the value of the integral is:

$I = \pi$

Would you like further details on any part of the solution? Here are some related questions to expand on this topic:

1. What other trigonometric substitutions are useful for similar integrals?
2. How do we handle improper integrals with more difficult boundary behavior?
3. Can we solve this integral using a different substitution method?
4. How does the convergence test change for more complex functions?
5. What are the implications of this result in probability or statistics?

Tip: Trigonometric substitutions are powerful tools when dealing with square roots of quadratic expressions in integrals.