To determine if the given integral is improper and to evaluate it, we need to analyze the integrand and the limits of integration.

Step 1: Determine if the Integral is Improper

An integral is considered improper if:

1. The limits of integration involve infinity.
2. The integrand becomes infinite at any point in the interval of integration.

The integral in question is:

$\int_{0}^{\frac{\pi}{4}} 10 \sin (2 t) \cos (2 t) \sqrt{\cos ^{2}(2 t) + 5} \, dt.$

Let's analyze the integrand:

$f(t) = 10 \sin (2t) \cos (2t) \sqrt{\cos^2(2t) + 5}.$

1. Limits of integration: The limits are from $0$ to $\frac{\pi}{4}$, which are finite.

2. Behavior of the integrand:

   • $\sin(2t)$ and $\cos(2t)$ are both bounded between $-1$ and $1$.
   • Therefore, $\sin(2t) \cos(2t)$ is also bounded between $-1$ and $1$.
   • The expression under the square root, $\cos^2(2t) + 5$, is always positive since $\cos^2(2t) \geq 0$ and the smallest value is $5$.
   • Hence, $\sqrt{\cos^2(2t) + 5}$ is well-defined and does not cause any discontinuity.

Since the integrand is continuous over the entire interval $[0, \frac{\pi}{4}]$, the integral is not improper.

Step 2: Evaluate the Integral

To evaluate the integral, we simplify it using trigonometric identities:

$\sin(2t) \cos(2t) = \frac{1}{2} \sin(4t).$

Thus, the integral becomes:

$\int_{0}^{\frac{\pi}{4}} 10 \sin (2 t) \cos (2 t) \sqrt{\cos ^{2}(2 t) + 5} \, dt = \int_{0}^{\frac{\pi}{4}} 10 \cdot \frac{1}{2} \sin(4t) \sqrt{\cos^2(2t) + 5} \, dt.$

Simplify it further:

$5 \int_{0}^{\frac{\pi}{4}} \sin(4t) \sqrt{\cos^2(2t) + 5} \, dt.$

This integral involves a composite function, making it non-trivial to evaluate directly. We can attempt a substitution or look for a more suitable method, such as integration by parts or using a table of integrals.

Substitution Approach

Let's set $u = \cos^2(2t) + 5$. Then:

$du = 2 \cos(2t) (-2 \sin(2t)) \, dt = -4 \sin(2t) \cos(2t) \, dt.$

This substitution does not simplify the integral in a straightforward manner. Given this complexity, it is better to leave the integral in its current form and recognize that the solution would likely involve advanced techniques or numerical methods.

Conclusion

1. Improperness: The integral is not improper because the integrand is continuous over the interval of integration.
2. Evaluation: The integral is complex and requires advanced techniques for a complete analytical solution.

Would you like a more detailed breakdown of the steps or another approach?

Here are 5 related questions to expand upon this topic:

1. What are the conditions for an integral to be considered improper?
2. How can substitution techniques simplify complex integrals?
3. What is the difference between definite and indefinite integrals in terms of improperness?
4. Are there any alternative methods to evaluate integrals that involve square roots and trigonometric functions?
5. How can numerical methods be applied to evaluate difficult integrals?

Tip: When dealing with trigonometric integrals, always check for possible identities or substitutions that can simplify the expression.