We are given that $G(x) = \int_{h_1(x)}^{h_2(x)} f(t) \, dt$, where:

• $h_1(x) = \pi \cos(2\pi x)$
• $h_2(x) = 4 \tan^{-1}(x)$
• $f(t) = \frac{188}{1 + t^2}$

We need to find $G(1) + G'(1)$.

Step 1: Find $G(1)$

To compute $G(1)$, we need to evaluate the definite integral for $x = 1$.

• $h_1(1) = \pi \cos(2\pi \cdot 1) = \pi \cos(2\pi) = \pi \cdot 1 = \pi$
• $h_2(1) = 4 \tan^{-1}(1) = 4 \cdot \frac{\pi}{4} = \pi$

Therefore, $G(1) = \int_{\pi}^{\pi} \frac{188}{1 + t^2} \, dt = 0$, because the limits of integration are equal.

Step 2: Differentiate $G(x)$ to find $G'(x)$

By the Leibniz rule for differentiating integrals with variable limits, we have: $G'(x) = f(h_2(x)) \cdot h_2'(x) - f(h_1(x)) \cdot h_1'(x)$ where $f(t) = \frac{188}{1 + t^2}$, $h_1(x) = \pi \cos(2\pi x)$, and $h_2(x) = 4 \tan^{-1}(x)$.

Step 2.1: Compute $h_1'(x)$ and $h_2'(x)$

• $h_1(x) = \pi \cos(2\pi x)$ $h_1'(x) = -2\pi^2 \sin(2\pi x)$

• $h_2(x) = 4 \tan^{-1}(x)$ $h_2'(x) = \frac{4}{1 + x^2}$

Step 2.2: Evaluate $G'(1)$

Now, evaluate each term at $x = 1$:

• $h_1(1) = \pi$
• $h_2(1) = \pi$
• $f(h_1(1)) = f(\pi) = \frac{188}{1 + \pi^2}$
• $f(h_2(1)) = f(\pi) = \frac{188}{1 + \pi^2}$
• $h_1'(1) = -2\pi^2 \sin(2\pi) = 0$
• $h_2'(1) = \frac{4}{1 + 1^2} = 2$

Thus, $G'(1) = f(\pi) \cdot 2 - f(\pi) \cdot 0 = 2 \cdot \frac{188}{1 + \pi^2}$

Step 3: Final Answer

Now we can compute $G(1) + G'(1)$:

$G(1) + G'(1) = 0 + 2 \cdot \frac{188}{1 + \pi^2} = \frac{376}{1 + \pi^2}$

This is the required result.

Would you like more details, or have any further questions?

Here are 5 related questions for practice:

1. How does the Leibniz rule apply to integrals with variable limits?
2. How would $G'(x)$ change if $h_1(x)$ and $h_2(x)$ were linear functions?
3. What is the geometric interpretation of the function $G(x)$?
4. How do you find the derivative of an inverse trigonometric function like $\tan^{-1}(x)$?
5. How does changing the function $f(t)$ affect the value of $G(1)$?

Tip: When differentiating definite integrals with variable limits, always remember to apply the Leibniz rule, accounting for both upper and lower limits of integration!