Ejercicios Suplementarios. 1. \( \int\left(3-x+x^{5}-2 x^{7}\right) d x \) 2. \( \int\left(5 \sqrt{x} \sqrt[3]{x}+\sqrt[4]{x}-\frac{2}{\sqrt[4]{x}}\right) d x \) 3. \( \int(2-2 x)^{3} x d x \) 4. \( \int \frac{3 x^{2}-4 x-4}{x-2} d x \) 5. \( \int \frac{(x+1)(x-2)}{\sqrt{x}} d x \)

Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(3x-x+x^{5}-2x^{7}\right) dx\) - step1: Subtract the terms: \(\int \left(2x+x^{5}-2x^{7}\right) dx\) - step2: Use properties of integrals: \(\int 2x dx+\int x^{5} dx+\int -2x^{7} dx\) - step3: Evaluate the integral: \(x^{2}+\int x^{5} dx+\int -2x^{7} dx\) - step4: Evaluate the integral: \(x^{2}+\frac{x^{6}}{6}+\int -2x^{7} dx\) - step5: Evaluate the integral: \(x^{2}+\frac{x^{6}}{6}-\frac{x^{8}}{4}\) - step6: Add the constant of integral C: \(x^{2}+\frac{x^{6}}{6}-\frac{x^{8}}{4} + C, C \in \mathbb{R}\) Calculate the integral \( (x+1)*(x-2)/\sqrt(x) \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \frac{\left(x+1\right)\left(x-2\right)}{\sqrt{x}} dx\) - step1: Multiply the terms: \(\int \frac{x^{2}-x-2}{\sqrt{x}} dx\) - step2: Rewrite the fraction: \(\int \left(x^{\frac{3}{2}}-x^{\frac{1}{2}}-\frac{2}{x^{\frac{1}{2}}}\right) dx\) - step3: Use properties of integrals: \(\int \left(x^{\frac{3}{2}}-x^{\frac{1}{2}}\right) dx+\int -\frac{2}{x^{\frac{1}{2}}} dx\) - step4: Evaluate the integral: \(\frac{2}{5}x^{\frac{5}{2}}-\frac{2}{3}x^{\frac{3}{2}}+\int -\frac{2}{x^{\frac{1}{2}}} dx\) - step5: Evaluate the integral: \(\frac{2}{5}x^{\frac{5}{2}}-\frac{2}{3}x^{\frac{3}{2}}-4x^{\frac{1}{2}}\) - step6: Simplify: \(\frac{2}{5}x^{2}\sqrt{x}-\frac{2}{3}x^{\frac{3}{2}}-4x^{\frac{1}{2}}\) - step7: Simplify: \(\frac{2}{5}x^{2}\sqrt{x}-\frac{2}{3}x\sqrt{x}-4x^{\frac{1}{2}}\) - step8: Simplify: \(\frac{2}{5}x^{2}\sqrt{x}-\frac{2}{3}x\sqrt{x}-4\sqrt{x}\) - step9: Add the constant of integral C: \(\frac{2}{5}x^{2}\sqrt{x}-\frac{2}{3}x\sqrt{x}-4\sqrt{x} + C, C \in \mathbb{R}\) Calculate the integral \( 5*\sqrt(x)*x^(1/3)+x^(1/4)-2/x^(1/4) \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \left(5\sqrt{x}\times x^{\frac{1}{3}}+x^{\frac{1}{4}}-\frac{2}{x^{\frac{1}{4}}}\right) dx\) - step1: Reorder the terms: \(\int \left(5\sqrt[6]{x^{5}}+x^{\frac{1}{4}}-\frac{2}{x^{\frac{1}{4}}}\right) dx\) - step2: Use properties of integrals: \(\int 5\sqrt[6]{x^{5}} dx+\int x^{\frac{1}{4}} dx+\int -\frac{2}{x^{\frac{1}{4}}} dx\) - step3: Evaluate the integral: \(\frac{30}{11}x^{\frac{11}{6}}+\int x^{\frac{1}{4}} dx+\int -\frac{2}{x^{\frac{1}{4}}} dx\) - step4: Evaluate the integral: \(\frac{30}{11}x^{\frac{11}{6}}+\frac{4}{5}x^{\frac{5}{4}}+\int -\frac{2}{x^{\frac{1}{4}}} dx\) - step5: Evaluate the integral: \(\frac{30}{11}x^{\frac{11}{6}}+\frac{4}{5}x^{\frac{5}{4}}-\frac{8}{3}x^{\frac{3}{4}}\) - step6: Simplify: \(\frac{30}{11}x\sqrt[6]{x^{5}}+\frac{4}{5}x^{\frac{5}{4}}-\frac{8}{3}x^{\frac{3}{4}}\) - step7: Simplify: \(\frac{30}{11}x\sqrt[6]{x^{5}}+\frac{4}{5}x\sqrt[4]{x}-\frac{8}{3}x^{\frac{3}{4}}\) - step8: Simplify: \(\frac{30}{11}x\sqrt[6]{x^{5}}+\frac{4}{5}x\sqrt[4]{x}-\frac{8}{3}\sqrt[4]{x^{3}}\) - step9: Add the constant of integral C: \(\frac{30}{11}x\sqrt[6]{x^{5}}+\frac{4}{5}x\sqrt[4]{x}-\frac{8}{3}\sqrt[4]{x^{3}} + C, C \in \mathbb{R}\) Calculate the integral \( (3*x^2-4*x-4)/(x-2) \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \frac{\left(3x^{2}-4x-4\right)}{\left(x-2\right)} dx\) - step1: Remove the parentheses: \(\int \frac{3x^{2}-4x-4}{x-2} dx\) - step2: Simplify the expression: \(\int \left(3x+2\right) dx\) - step3: Use properties of integrals: \(\int 3x dx+\int 2 dx\) - step4: Evaluate the integral: \(\frac{3x^{2}}{2}+\int 2 dx\) - step5: Evaluate the integral: \(\frac{3x^{2}}{2}+2x\) - step6: Add the constant of integral C: \(\frac{3x^{2}}{2}+2x + C, C \in \mathbb{R}\) Calculate the integral \( (2-2*x)^3*x \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \left(2-2x\right)^{3}x dx\) - step1: Use the substitution \(dx=-\frac{1}{2} dt\) to transform the integral\(:\) \(\int \left(2-2x\right)^{3}x\left(-\frac{1}{2}\right) dt\) - step2: Simplify: \(\int -\frac{\left(2-2x\right)^{3}}{2}x dt\) - step3: Use the substitution \(t=2-2x\) to transform the integral\(:\) \(\int \frac{-t^{3}}{2}\times \left(-\frac{1}{2}t+1\right) dt\) - step4: Simplify: \(\int \frac{\frac{1}{2}t^{4}-t^{3}}{2} dt\) - step5: Rewrite the expression: \(\int \frac{1}{2}\left(\frac{1}{2}t^{4}-t^{3}\right) dt\) - step6: Use properties of integrals: \(\frac{1}{2}\times \int \left(\frac{1}{2}t^{4}-t^{3}\right) dt\) - step7: Use properties of integrals: \(\frac{1}{2}\left(\int \frac{1}{2}t^{4} dt+\int -t^{3} dt\right)\) - step8: Calculate: \(\frac{1}{2}\times \int \frac{1}{2}t^{4} dt+\frac{1}{2}\times \int -t^{3} dt\) - step9: Evaluate the integral: \(\frac{t^{5}}{20}+\frac{1}{2}\times \int -t^{3} dt\) - step10: Evaluate the integral: \(\frac{t^{5}}{20}-\frac{t^{4}}{8}\) - step11: Substitute back: \(\frac{\left(2-2x\right)^{5}}{20}-\frac{\left(2-2x\right)^{4}}{8}\) - step12: Rewrite the expression: \(\frac{8\left(1-x\right)^{5}}{5}-\frac{\left(2-2x\right)^{4}}{8}\) - step13: Rewrite the expression: \(\frac{8\left(1-x\right)^{5}}{5}-2\left(1-x\right)^{4}\) - step14: Calculate: \(\frac{8\left(1-x\right)^{5}}{5}-2+8x-12x^{2}+8x^{3}-2x^{4}\) - step15: Expand the expression: \(\frac{8-40x+80x^{2}-80x^{3}+40x^{4}-8x^{5}}{5}-2+8x-12x^{2}+8x^{3}-2x^{4}\) - step16: Add the constant of integral C: \(\frac{8-40x+80x^{2}-80x^{3}+40x^{4}-8x^{5}}{5}-2+8x-12x^{2}+8x^{3}-2x^{4} + C, C \in \mathbb{R}\) Aquí están las soluciones a los ejercicios de integración: 1. Para la integral \( \int(3-x+x^{5}-2 x^{7}) d x \): \[ \int(3-x+x^{5}-2 x^{7}) d x = x^{2}+\frac{x^{6}}{6}-\frac{x^{8}}{4} + C, \quad C \in \mathbb{R} \] 2. Para la integral \( \int\left(5 \sqrt{x} \sqrt[3]{x}+\sqrt[4]{x}-\frac{2}{\sqrt[4]{x}}\right) d x \): \[ \int\left(5 \sqrt{x} \sqrt[3]{x}+\sqrt[4]{x}-\frac{2}{\sqrt[4]{x}}\right) d x = \frac{30}{11}x\sqrt[6]{x^{5}}+\frac{4}{5}x\sqrt[4]{x}-\frac{8}{3}\sqrt[4]{x^{3}} + C, \quad C \in \mathbb{R} \] 3. Para la integral \( \int(2-2 x)^{3} x d x \): \[ \int(2-2 x)^{3} x d x = \frac{8-40x+80x^{2}-80x^{3}+40x^{4}-8x^{5}}{5}-2+8x-12x^{2}+8x^{3}-2x^{4} + C, \quad C \in \mathbb{R} \] 4. Para la integral \( \int \frac{3 x^{2}-4 x-4}{x-2} d x \): \[ \int \frac{3 x^{2}-4 x-4}{x-2} d x = \frac{3x^{2}}{2}+2x + C, \quad C \in \mathbb{R} \] 5. Para la integral \( \int \frac{(x+1)(x-2)}{\sqrt{x}} d x \): \[ \int \frac{(x+1)(x-2)}{\sqrt{x}} d x = \frac{2}{5}x^{2}\sqrt{x}-\frac{2}{3}x\sqrt{x}-4\sqrt{x} + C, \quad C \in \mathbb{R} \] Si necesitas más ayuda o aclaraciones sobre algún ejercicio, no dudes en preguntar.