1. Problem 9: Find the functions $f(x)$ given their derivatives $f'(x)$. 2. To find $f(x)$, integrate each derivative function with respect to $x$. Remember to add a constant of integration $C$. 9.a. Given: $f'(x) = 4x^3$ Integrate: $$f(x) = \int 4x^3 \, dx = 4 \cdot \frac{x^{4}}{4} + C = x^4 + C$$ 9.б. Given: $f'(x) = 3x^2 - 4x^5$ Integrate term-wise: $$f(x) = \int (3x^2 - 4x^5) \, dx = 3 \cdot \frac{x^3}{3} - 4 \cdot \frac{x^6}{6} + C = x^3 - \frac{2}{3} x^6 + C$$ 9.в. Given: $f'(x) = 9x^8 + 7x^6 + 1$ Integrate: $$f(x) = 9 \cdot \frac{x^9}{9} + 7 \cdot \frac{x^7}{7} + x + C = x^9 + x^7 + x + C$$ 9.г. Given: $f'(x) = -10x^9 + 8x^7 - 2x$ Integrate: $$f(x) = -10 \cdot \frac{x^{10}}{10} + 8 \cdot \frac{x^8}{8} - 2 \cdot \frac{x^2}{2} + C = -x^{10} + x^8 - x^2 + C$$ 9.д. Given: $f'(x) = 2x - 4x^3 + 7$ Integrate: $$f(x) = 2 \cdot \frac{x^2}{2} - 4 \cdot \frac{x^4}{4} + 7x + C = x^2 - x^4 + 7x + C$$ 9.е. Given: $f'(x) = 7x^6 - 6x^5 + 5x^4$ Integrate: $$f(x) = 7 \cdot \frac{x^7}{7} - 6 \cdot \frac{x^6}{6} + 5 \cdot \frac{x^5}{5} + C = x^7 - x^6 + x^5 + C$$ 9.ё. Given: $f'(x) = \frac{1}{2}x - \frac{1}{3}x^2 + \frac{1}{4}$ Integrate: $$f(x) = \frac{1}{2} \cdot \frac{x^2}{2} - \frac{1}{3} \cdot \frac{x^3}{3} + \frac{1}{4}x + C = \frac{1}{4}x^2 - \frac{1}{9}x^3 + \frac{1}{4}x + C$$ 9.ж. Given: $f'(x) = 6x^{-4} - 4x^{-3}$ Integrate: $$f(x) = 6 \cdot \frac{x^{-3}}{-3} - 4 \cdot \frac{x^{-2}}{-2} + C = -2x^{-3} + 2x^{-2} + C$$ 9.з. Given: $f'(x) = 3x^{4} - 8x^{\frac{4}{3}} + 2$ Integrate: $$f(x) = 3 \cdot \frac{x^5}{5} - 8 \cdot \frac{x^{\frac{7}{3}}}{\frac{7}{3}} + 2x + C = \frac{3}{5}x^5 - \frac{24}{7}x^{\frac{7}{3}} + 2x + C$$ 10. Problem: Find $y(x)$ given $\frac{dy}{dx}$. 10.a. $\frac{dy}{dx} = x^3 + x^2 + 2$ Integrate: $$y = \int (x^3 + x^2 + 2)dx = \frac{x^4}{4} + \frac{x^3}{3} + 2x + C$$ 10.б. $\frac{dy}{dx} = 4x^2 + x - 5$ Integrate: $$y = \frac{4x^3}{3} + \frac{x^2}{2} - 5x + C$$ 10.в. $\frac{dy}{dx} = \frac{2}{3}x^3 - \frac{1}{2}x^2 + \frac{1}{3}x - \frac{1}{2}$ Integrate: $$y = \frac{2}{3} \cdot \frac{x^4}{4} - \frac{1}{2} \cdot \frac{x^3}{3} + \frac{1}{3} \cdot \frac{x^2}{2} - \frac{1}{2}x + C = \frac{1}{6}x^4 - \frac{1}{6}x^3 + \frac{1}{6}x^2 - \frac{1}{2}x + C$$ 10.г. $\frac{dy}{dx} = x^3 + x^2 + x + 1$ Integrate: $$y = \frac{x^4}{4} + \frac{x^3}{3} + \frac{x^2}{2} + x + C$$ 10.д. $\frac{dy}{dx} = 2\sqrt{x} + 3\sqrt[3]{x} = 2x^{1/2} + 3x^{1/3}$ Integrate: $$y = 2 \cdot \frac{x^{3/2}}{\frac{3}{2}} + 3 \cdot \frac{x^{4/3}}{\frac{4}{3}} + C = \frac{4}{3}x^{3/2} + \frac{9}{4}x^{4/3} + C$$ 10.е. $\frac{dy}{dx} = \frac{4}{\sqrt[3]{x}} = 4x^{-1/3}$ Integrate: $$y = 4 \cdot \frac{x^{2/3}}{\frac{2}{3}} + C = 6x^{2/3} + C$$ 10.ё. $\frac{dy}{dx} = \frac{x + 6}{\sqrt{x}} = \frac{x}{x^{1/2}} + \frac{6}{x^{1/2}} = x^{1/2} + 6x^{-1/2}$ Integrate: $$y = \frac{2}{3}x^{3/2} + 12x^{1/2} + C$$ Final answers include constants of integration $C$ in each case.