To determine whether the given integral is improper or proper, we need to check the integrand for any potential issues at the limits of integration or within the interval.

The given integral is:

$\int_0^{\frac{\pi}{4}} \left( 10 \sin(2t) \cos(2t) \sqrt{\cos^2(2t)} + 5 \right) dt$

1. Checking for discontinuities or undefined points:

• The integrand is composed of trigonometric functions and a square root function.
• $\sin(2t)$ and $\cos(2t)$ are smooth and continuous for all $t$.
• $\cos^2(2t)$ is always non-negative, and the square root of $\cos^2(2t)$ is simply $|\cos(2t)|$, which is continuous and finite in the interval $[0, \frac{\pi}{4}]$.
• The constant term $5$ does not cause any issues.

There are no discontinuities or points where the integrand becomes infinite or undefined within the given interval $[0, \frac{\pi}{4}]$.

2. Checking the limits of integration:

• The lower limit $t = 0$ and the upper limit $t = \frac{\pi}{4}$ do not introduce any issues, as the integrand remains finite at both points.

Conclusion:

This is a proper integral because the integrand is continuous and finite over the entire interval of integration $[0, \frac{\pi}{4}]$.

Would you like to proceed with the evaluation of the integral, or do you have any questions about this step?

Here are 5 related questions to explore:

1. What is the difference between proper and improper integrals?
2. How do you handle improper integrals if there is a discontinuity at a limit?
3. What other types of singularities can make an integral improper?
4. Can you give examples of improper integrals in real-world applications?
5. How do you compute the limit of an improper integral if the integrand has a vertical asymptote?

Tip: Always examine the integrand for points of discontinuity or infinite behavior when determining if an integral is proper or improper.