1. Encuentra la antiderivada de las siguientes funciones aplicando la regla indicada. 1. \( f(x)=x^{9} \) 2. \( f(x)=2 x^{2 / 3}-5 x^{3} \) 3. \( f(x)=-3 x^{-4}+10 x^{2 / 3}-7 x \) 4. \( f(x)=\frac{x}{2}-4 x^{3}-x^{4} \) 5. \( f(x)=3 x+8 x^{3}+x^{-3}+5 x^{2} \) 6. \( f(x)=9 x^{4 / 5}+x-\frac{2}{x^{3}}-\sqrt{x} \) 7. \( f(x)=\frac{x^{3}}{2}-3 x^{-2}-2 x^{3}-x^{3 / 2} \) 8. \( f(x)=2 x^{3}-4 x^{2}+2 x \); considere que \( F(0)=15 \) 9. \( g(x)=x^{2}+2 x-1 ; \) considereque \( G(1)=7 \) 10. \( f(x)=x^{3}-3 x^{2}+6 x \); considere que \( F(-1)=3 \)

Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int x^{9} dx\) - step1: Evaluate the integral: \(\frac{x^{9+1}}{9+1}\) - step2: Add the numbers: \(\frac{x^{10}}{9+1}\) - step3: Add the numbers: \(\frac{x^{10}}{10}\) - step4: Add the constant of integral C: \(\frac{x^{10}}{10} + C, C \in \mathbb{R}\) Calculate the integral \( x^{2}+2 x-1 \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(x^{2}+2x-1\right) dx\) - step1: Use properties of integrals: \(\int x^{2} dx+\int 2x dx+\int -1 dx\) - step2: Evaluate the integral: \(\frac{x^{3}}{3}+\int 2x dx+\int -1 dx\) - step3: Evaluate the integral: \(\frac{x^{3}}{3}+x^{2}+\int -1 dx\) - step4: Evaluate the integral: \(\frac{x^{3}}{3}+x^{2}-x\) - step5: Add the constant of integral C: \(\frac{x^{3}}{3}+x^{2}-x + C, C \in \mathbb{R}\) Calculate the integral \( x^{3}-3 x^{2}+6 x \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(x^{3}-3x^{2}+6x\right) dx\) - step1: Use properties of integrals: \(\int x^{3} dx+\int -3x^{2} dx+\int 6x dx\) - step2: Evaluate the integral: \(\frac{x^{4}}{4}+\int -3x^{2} dx+\int 6x dx\) - step3: Evaluate the integral: \(\frac{x^{4}}{4}-x^{3}+\int 6x dx\) - step4: Evaluate the integral: \(\frac{x^{4}}{4}-x^{3}+3x^{2}\) - step5: Add the constant of integral C: \(\frac{x^{4}}{4}-x^{3}+3x^{2} + C, C \in \mathbb{R}\) Calculate the integral \( 2 x^{3}-4 x^{2}+2 x \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(2x^{3}-4x^{2}+2x\right) dx\) - step1: Use properties of integrals: \(\int 2x^{3} dx+\int -4x^{2} dx+\int 2x dx\) - step2: Evaluate the integral: \(\frac{x^{4}}{2}+\int -4x^{2} dx+\int 2x dx\) - step3: Evaluate the integral: \(\frac{x^{4}}{2}-\frac{4x^{3}}{3}+\int 2x dx\) - step4: Evaluate the integral: \(\frac{x^{4}}{2}-\frac{4x^{3}}{3}+x^{2}\) - step5: Add the constant of integral C: \(\frac{x^{4}}{2}-\frac{4x^{3}}{3}+x^{2} + C, C \in \mathbb{R}\) Calculate the integral \( \frac{x}{2}-4 x^{3}-x^{4} \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(\frac{x}{2}-4x^{3}-x^{4}\right) dx\) - step1: Use properties of integrals: \(\int \frac{x}{2} dx+\int -4x^{3} dx+\int -x^{4} dx\) - step2: Evaluate the integral: \(\frac{x^{2}}{4}+\int -4x^{3} dx+\int -x^{4} dx\) - step3: Evaluate the integral: \(\frac{x^{2}}{4}-x^{4}+\int -x^{4} dx\) - step4: Evaluate the integral: \(\frac{x^{2}}{4}-x^{4}-\frac{x^{5}}{5}\) - step5: Add the constant of integral C: \(\frac{x^{2}}{4}-x^{4}-\frac{x^{5}}{5} + C, C \in \mathbb{R}\) Calculate the integral \( 2 x^{2 / 3}-5 x^{3} \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(2x^{\frac{2}{3}}-5x^{3}\right) dx\) - step1: Use properties of integrals: \(\int 2x^{\frac{2}{3}} dx+\int -5x^{3} dx\) - step2: Evaluate the integral: \(\frac{6}{5}x^{\frac{5}{3}}+\int -5x^{3} dx\) - step3: Evaluate the integral: \(\frac{6}{5}x^{\frac{5}{3}}-\frac{5x^{4}}{4}\) - step4: Simplify: \(\frac{6}{5}x\sqrt[3]{x^{2}}-\frac{5x^{4}}{4}\) - step5: Add the constant of integral C: \(\frac{6}{5}x\sqrt[3]{x^{2}}-\frac{5x^{4}}{4} + C, C \in \mathbb{R}\) Calculate the integral \( 3 x+8 x^{3}+x^{-3}+5 x^{2} \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(3x+8x^{3}+x^{-3}+5x^{2}\right) dx\) - step1: Use properties of integrals: \(\int 3x dx+\int 8x^{3} dx+\int x^{-3} dx+\int 5x^{2} dx\) - step2: Evaluate the integral: \(\frac{3x^{2}}{2}+\int 8x^{3} dx+\int x^{-3} dx+\int 5x^{2} dx\) - step3: Evaluate the integral: \(\frac{3x^{2}}{2}+2x^{4}+\int x^{-3} dx+\int 5x^{2} dx\) - step4: Evaluate the integral: \(\frac{3x^{2}}{2}+2x^{4}-\frac{1}{2x^{2}}+\int 5x^{2} dx\) - step5: Evaluate the integral: \(\frac{3x^{2}}{2}+2x^{4}-\frac{1}{2x^{2}}+\frac{5x^{3}}{3}\) - step6: Add the constant of integral C: \(\frac{3x^{2}}{2}+2x^{4}-\frac{1}{2x^{2}}+\frac{5x^{3}}{3} + C, C \in \mathbb{R}\) Calculate the integral \( -3 x^{-4}+10 x^{2 / 3}-7 x \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(-3x^{-4}+10x^{\frac{2}{3}}-7x\right) dx\) - step1: Use properties of integrals: \(\int -3x^{-4} dx+\int 10x^{\frac{2}{3}} dx+\int -7x dx\) - step2: Evaluate the integral: \(\frac{1}{x^{3}}+\int 10x^{\frac{2}{3}} dx+\int -7x dx\) - step3: Evaluate the integral: \(\frac{1}{x^{3}}+6x^{\frac{5}{3}}+\int -7x dx\) - step4: Evaluate the integral: \(\frac{1}{x^{3}}+6x^{\frac{5}{3}}-\frac{7x^{2}}{2}\) - step5: Simplify: \(\frac{1}{x^{3}}+6x\sqrt[3]{x^{2}}-\frac{7x^{2}}{2}\) - step6: Add the constant of integral C: \(\frac{1}{x^{3}}+6x\sqrt[3]{x^{2}}-\frac{7x^{2}}{2} + C, C \in \mathbb{R}\) Calculate the integral \( \frac{x^{3}}{2}-3 x^{-2}-2 x^{3}-x^{3 / 2} \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(\frac{x^{3}}{2}-3x^{-2}-2x^{3}-x^{\frac{3}{2}}\right) dx\) - step1: Use properties of integrals: \(\int \frac{x^{3}}{2} dx+\int -3x^{-2} dx+\int -2x^{3} dx+\int -x^{\frac{3}{2}} dx\) - step2: Evaluate the integral: \(\frac{x^{4}}{8}+\int -3x^{-2} dx+\int -2x^{3} dx+\int -x^{\frac{3}{2}} dx\) - step3: Evaluate the integral: \(\frac{x^{4}}{8}+\frac{3}{x}+\int -2x^{3} dx+\int -x^{\frac{3}{2}} dx\) - step4: Evaluate the integral: \(\frac{x^{4}}{8}+\frac{3}{x}-\frac{x^{4}}{2}+\int -x^{\frac{3}{2}} dx\) - step5: Evaluate the integral: \(\frac{x^{4}}{8}+\frac{3}{x}-\frac{x^{4}}{2}-\frac{2}{5}x^{\frac{5}{2}}\) - step6: Simplify: \(\frac{x^{4}}{8}+\frac{3}{x}-\frac{x^{4}}{2}-\frac{2}{5}x^{2}\sqrt{x}\) - step7: Add the constant of integral C: \(\frac{x^{4}}{8}+\frac{3}{x}-\frac{x^{4}}{2}-\frac{2}{5}x^{2}\sqrt{x} + C, C \in \mathbb{R}\) Calculate the integral \( 9 x^{4 / 5}+x-\frac{2}{x^{3}}-\sqrt{x} \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(9x^{\frac{4}{5}}+x-\frac{2}{x^{3}}-\sqrt{x}\right) dx\) - step1: Use properties of integrals: \(\int 9x^{\frac{4}{5}} dx+\int x dx+\int -\frac{2}{x^{3}} dx+\int -\sqrt{x} dx\) - step2: Evaluate the integral: \(5x^{\frac{9}{5}}+\int x dx+\int -\frac{2}{x^{3}} dx+\int -\sqrt{x} dx\) - step3: Evaluate the integral: \(5x^{\frac{9}{5}}+\frac{x^{2}}{2}+\int -\frac{2}{x^{3}} dx+\int -\sqrt{x} dx\) - step4: Evaluate the integral: \(5x^{\frac{9}{5}}+\frac{x^{2}}{2}+\frac{1}{x^{2}}+\int -\sqrt{x} dx\) - step5: Evaluate the integral: \(5x^{\frac{9}{5}}+\frac{x^{2}}{2}+\frac{1}{x^{2}}-\frac{2}{3}x^{\frac{3}{2}}\) - step6: Simplify: \(5x\sqrt[5]{x^{4}}+\frac{x^{2}}{2}+\frac{1}{x^{2}}-\frac{2}{3}x^{\frac{3}{2}}\) - step7: Simplify: \(5x\sqrt[5]{x^{4}}+\frac{x^{2}}{2}+\frac{1}{x^{2}}-\frac{2}{3}x\sqrt{x}\) - step8: Add the constant of integral C: \(5x\sqrt[5]{x^{4}}+\frac{x^{2}}{2}+\frac{1}{x^{2}}-\frac{2}{3}x\sqrt{x} + C, C \in \mathbb{R}\) Aquí están las antiderivadas de las funciones dadas: 1. Para \( f(x) = x^{9} \): \[ F(x) = \frac{x^{10}}{10} + C, \quad C \in \mathbb{R} \] 2. Para \( f(x) = 2 x^{2/3} - 5 x^{3} \): \[ F(x) = \frac{6}{5}x\sqrt[3]{x^{2}} - \frac{5x^{4}}{4} + C, \quad C \in \mathbb{R} \] 3. Para \( f(x) = -3 x^{-4} + 10 x^{2/3} - 7 x \): \[ F(x) = \frac{1}{x^{3}} + 6x\sqrt[3]{x^{2}} - \frac{7x^{2}}{2} + C, \quad C \in \mathbb{R} \] 4. Para \( f(x) = \frac{x}{2} - 4 x^{3} - x^{4} \): \[ F(x) = \frac{x^{2}}{4} - x^{4} - \frac{x^{5}}{5} + C, \quad C \in \mathbb{R} \] 5. Para \( f(x) = 3 x + 8 x^{3} + x^{-3} + 5 x^{2} \): \[ F(x) = \frac{3x^{2}}{2} + 2x^{4} - \frac{1}{2x^{2}} + \frac{5x^{3}}{3} + C, \quad C \in \mathbb{R} \] 6. Para \( f(x) = 9 x^{4/5} + x - \frac{2}{x^{3}} - \sqrt{x} \): \[ F(x) = 5x\sqrt[5]{x^{4}} + \frac{x^{2}}{2} + \frac{1}{x^{2}} - \frac{2}{3}x\sqrt{x} + C, \quad C \in \mathbb{R} \] 7. Para \( f(x) = \frac{x^{3}}{2} - 3 x^{-2} - 2 x^{3} - x^{3/2} \): \[ F(x) = \frac{x^{4}}{8} + \frac{3}{x} - \frac{x^{4}}{2} - \frac{2}{5}x^{2}\sqrt{x} + C, \quad C \in \mathbb{R} \] 8. Para \( f(x) = 2 x^{3} - 4 x^{2} + 2 x \); considerando que \( F(0) = 15 \): \[ F(x) = \frac{x^{4}}{2} - \frac{4x^{3}}{3} + x^{2} + C, \quad C \text{ se determina usando } F(0) = 15 \] 9. Para \( g(x) = x^{2} + 2 x - 1 \); considerando que \( G(1) = 7 \): \[ G(x) = \frac{x^{3}}{3} + x^{2} - x + C, \quad C \text{ se determina usando } G(1) = 7 \] 10. Para \( f(x) = x^{3} - 3 x^{2} + 6 x \); considerando que \( F(-1) = 3 \): \[ F(x) = \frac{x^{4}}{4} - x^{3} + 3x^{2} + C, \quad C \text{ se determina usando } F(-1) = 3 \] Si necesitas que calculemos los valores de \( C \) para los casos 8, 9 y 10, házmelo saber.