We are asked to determine whether the given integral is improper and, if not, to evaluate it.

Problem:

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

Step 1: Checking if the Integral is Improper

An integral is called improper if:

1. The limits of integration are infinite.
2. The integrand has a discontinuity or tends to infinity within the integration limits.

In this case, the limits of integration are $0$ to $\frac{\pi}{4}$, both of which are finite. Therefore, there is no issue with infinite limits.

Next, let's consider the integrand: $f(t) = 10 \sin(2t) \cos(2t) \sqrt{\cos^2(2t) + 5}$

• The function $\sin(2t)$ and $\cos(2t)$ are continuous and well-defined for all $t$.
• The term $\sqrt{\cos^2(2t) + 5}$ is always positive and continuous for all $t$ because $\cos^2(2t) + 5 \geq 5$, so there is no concern of division by zero or taking the square root of a negative number.

Thus, the integral is not improper.

Step 2: Simplifying the Integrand

We can simplify the product $\sin(2t) \cos(2t)$ using a trigonometric identity: $\sin(2t) \cos(2t) = \frac{1}{2} \sin(4t)$ So the integrand becomes: $f(t) = 5 \sin(4t) \sqrt{\cos^2(2t) + 5}$

Step 3: Evaluating the Integral

The integral becomes: $I = \int_0^{\frac{\pi}{4}} 5 \sin(4t) \sqrt{\cos^2(2t) + 5} \, dt$ This is a non-trivial integral and does not have an elementary antiderivative. Therefore, this integral would likely require numerical methods or special functions for evaluation.

Would you like me to proceed with a numerical evaluation for the integral, or would you prefer another approach?

Do you have any questions or need further clarification? Here are five related questions you could explore:

1. What makes an integral improper in general?
2. How can we simplify other trigonometric integrals?
3. Can this type of integral be evaluated using numerical methods like Simpson's rule?
4. What are other examples of improper integrals?
5. How does the trigonometric identity used here simplify integral expressions?

Tip: Always check for discontinuities or singularities in the integrand when determining if an integral is improper.