To evaluate the integral:

$I = \int_0^{\frac{\pi}{2}} \frac{1}{1 + \cos^2 x} \, dx$

we'll use a combination of trigonometric identities and methods of integration. Let’s break it down step by step.

Step 1: Simplify the integrand

The integrand is $\frac{1}{1 + \cos^2 x}$, which doesn’t directly suggest a simple elementary antiderivative. A common approach is to use a standard trigonometric substitution or to look for a known result. One useful identity for integrals of this type involves the tangent half-angle substitution, also called the Weierstrass substitution:

$t = \tan \left( \frac{x}{2} \right)$ This substitution transforms the trigonometric functions into rational functions of $t$, simplifying the integral. The relationships for this substitution are:

• $\cos x = \frac{1 - t^2}{1 + t^2}$
• $dx = \frac{2}{1 + t^2} \, dt$

Step 2: Apply the substitution

Using the Weierstrass substitution, let’s change the limits and rewrite the integrand in terms of $t$.

• When $x = 0$, $t = \tan(0) = 0$.
• When $x = \frac{\pi}{2}$, $t = \tan\left( \frac{\pi}{4} \right) = 1$.

The integral becomes:

$I = \int_0^1 \frac{2}{(1 + t^2)\left( 1 + \frac{1 - t^2}{1 + t^2} \right)} \, dt$

Simplifying the expression inside the integral:

$1 + \frac{1 - t^2}{1 + t^2} = \frac{1 + t^2 + 1 - t^2}{1 + t^2} = \frac{2}{1 + t^2}$

So the integral becomes:

$I = \int_0^1 \frac{2}{(1 + t^2) \cdot \frac{2}{1 + t^2}} \, dt$

This simplifies to:

$I = \int_0^1 1 \, dt = t \Big|_0^1 = 1 - 0 = 1$

Therefore, the value of the integral is $I = 1$.

Step 3: Final Answer

The value of the integral is:

$\boxed{1}$

It appears there may be a mistake in the original result provided (since you stated the result as $\frac{\pi}{2\sqrt{2}}$). Based on the standard methods and simplifications, the correct evaluation of this integral is $I = 1$.

Would you like further details or clarification on any part of the solution?

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Here are 5 related questions you could explore:

1. How do you derive the Weierstrass substitution in trigonometric integrals?
2. Can you evaluate the integral of $\frac{1}{1 + \sin^2 x}$?
3. What other methods can be used for integrals involving trigonometric functions like $\cos^2 x$?
4. How do you apply the substitution $t = \tan\left( \frac{x}{2} \right)$ to integrals of other trigonometric functions?
5. How do you use symmetry in integrals involving trigonometric functions?

Tip: For integrals involving trigonometric functions, always check for standard substitutions such as the Weierstrass substitution or look for symmetry in the bounds that might simplify the process.