1. **Problem 5:** Evaluate $$\frac{d}{dx} \int_1^{5x} e^{t^2} dt + \int_1^x \frac{d}{dx} \left[ \sqrt{1 + \ln(x)} \right] dx + \frac{d}{dx} \int_{\pi}^{2\pi} \frac{\sin x}{x} dx.$$ 2. **Step 1: Differentiate the first term using the Leibniz rule** $$\frac{d}{dx} \int_1^{5x} e^{t^2} dt = e^{(5x)^2} \cdot \frac{d}{dx}(5x) = e^{25x^2} \cdot 5 = 5 e^{25x^2}.$$ 3. **Step 2: Simplify the second term** Inside the integral, the derivative with respect to $x$ of $\sqrt{1 + \ln(x)}$ is $$\frac{d}{dx} \sqrt{1 + \ln(x)} = \frac{1}{2\sqrt{1 + \ln(x)}} \cdot \frac{1}{x} = \frac{1}{2x \sqrt{1 + \ln(x)}}.$$ 4. **Step 3: Evaluate the integral of the derivative from 1 to $x$** $$\int_1^x \frac{1}{2t \sqrt{1 + \ln(t)}} dt.$$ This integral is exactly the difference of the original function evaluated at $x$ and 1, by the Fundamental Theorem of Calculus: $$\int_1^x \frac{d}{dt} \sqrt{1 + \ln(t)} dt = \sqrt{1 + \ln(x)} - \sqrt{1 + \ln(1)} = \sqrt{1 + \ln(x)} - 1.$$ 5. **Step 4: Differentiate the third term** Since the limits of integration are constants, the derivative of $$\int_{\pi}^{2\pi} \frac{\sin x}{x} dx$$ is zero: $$\frac{d}{dx} \int_{\pi}^{2\pi} \frac{\sin x}{x} dx = 0.$$ 6. **Step 5: Combine all parts** $$5 e^{25x^2} + \left( \sqrt{1 + \ln(x)} - 1 \right) + 0 = 5 e^{25x^2} + \sqrt{1 + \ln(x)} - 1.$$ --- 7. **Problem 6:** Given $$f(x) = \int_{4x}^{6x} g(t) dt, \quad g(x) = \int_1^{3x} \sin t dt,$$ find $f''(x)$. 8. **Step 1: Find $g(x)$ and $g'(x)$** By the Fundamental Theorem of Calculus and chain rule: $$g(x) = \int_1^{3x} \sin t dt,$$ so $$g'(x) = \sin(3x) \cdot 3 = 3 \sin(3x).$$ 9. **Step 2: Differentiate $f(x)$ using Leibniz rule** $$f(x) = \int_{4x}^{6x} g(t) dt,$$ so $$f'(x) = g(6x) \cdot 6 - g(4x) \cdot 4.$$ 10. **Step 3: Differentiate $f'(x)$ to find $f''(x)$** $$f''(x) = \frac{d}{dx} \left[ 6 g(6x) - 4 g(4x) \right] = 6 \cdot g'(6x) \cdot 6 - 4 \cdot g'(4x) \cdot 4 = 36 g'(6x) - 16 g'(4x).$$ 11. **Step 4: Substitute $g'(x)$** $$f''(x) = 36 \cdot 3 \sin(18x) - 16 \cdot 3 \sin(12x) = 108 \sin(18x) - 48 \sin(12x).$$ **Final answers:** $$\boxed{5 e^{25x^2} + \sqrt{1 + \ln(x)} - 1}$$ and $$\boxed{f''(x) = 108 \sin(18x) - 48 \sin(12x)}.$$