Let's break down and solve the given integral:

The integral is:

$\frac{1}{4} \int_{12}^{16} \left( 61 + 10 \sin\left( \frac{\pi(t - 11)}{12} \right) \right) \, dt$

Step 1: Split the integral

We can split the integral into two parts:

$\frac{1}{4} \int_{12}^{16} \left( 61 \right) \, dt + \frac{1}{4} \int_{12}^{16} \left( 10 \sin\left( \frac{\pi(t - 11)}{12} \right) \right) \, dt$

Step 2: Evaluate the first part

The first integral is:

$\frac{1}{4} \int_{12}^{16} 61 \, dt$

Since $61$ is a constant:

$\frac{1}{4} \times 61 \times (16 - 12) = \frac{1}{4} \times 61 \times 4 = 61$

Step 3: Evaluate the second part

The second integral is:

$\frac{1}{4} \int_{12}^{16} 10 \sin\left( \frac{\pi(t - 11)}{12} \right) \, dt$

First, take the constant $10$ outside the integral:

$\frac{10}{4} \int_{12}^{16} \sin\left( \frac{\pi(t - 11)}{12} \right) \, dt = \frac{5}{2} \int_{12}^{16} \sin\left( \frac{\pi(t - 11)}{12} \right) \, dt$

Now, let’s perform a substitution. Set:

$u = \frac{\pi(t - 11)}{12} \quad \text{so} \quad du = \frac{\pi}{12} dt \quad \text{or} \quad dt = \frac{12}{\pi} du$

When $t = 12$, $u = \frac{\pi}{12}$, and when $t = 16$, $u = \frac{5\pi}{12}$.

Thus, the integral becomes:

$\frac{5}{2} \times \frac{12}{\pi} \int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \sin(u) \, du$

The integral of $\sin(u)$ is $-\cos(u)$:

[ \frac{5}{2} \times \frac{12}{\pi} \left[ -\cos(u) \right]_{\frac{\pi}{12}}^{\frac{5\pi}{12}} = -\frac{30}{\pi} \left[ \cos\left( \frac{5\pi}{12} \right) - \cos\left( \frac{\pi}{12} \right) \right] ]

Step 4: Compute the values of cosine

We can now compute the values of $\cos\left( \frac{5\pi}{12} \right)$ and $\cos\left( \frac{\pi}{12} \right)$. Using known trigonometric values:

$\cos\left( \frac{5\pi}{12} \right) \approx -0.2588, \quad \cos\left( \frac{\pi}{12} \right) \approx 0.9659$

Thus, the difference is:

$\cos\left( \frac{5\pi}{12} \right) - \cos\left( \frac{\pi}{12} \right) = -0.2588 - 0.9659 = -1.2247$

Step 5: Final calculation

Now substitute this back into the expression:

$-\frac{30}{\pi} \times (-1.2247) \approx \frac{30 \times 1.2247}{3.1416} \approx \frac{36.741}{3.1416} \approx 11.7$

Step 6: Combine both results

Now, adding the two parts together:

$61 + 11.7 = 72.7$

Thus, the value of the integral is approximately 72.7.

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Would you like more details on any specific step? Here are some follow-up questions that might help you understand this better:

1. How does the substitution method work in integrals?
2. What is the role of trigonometric identities in integral calculations?
3. Why do we separate constants from variable terms in an integral?
4. Can integrals always be solved using substitution?
5. How can we calculate cosine values without a calculator?

Tip: Remember, substitution is a powerful technique for simplifying integrals when the argument of a function is complex.