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. $$ \int \frac{(x+1)(x-2)}{\sqrt{x}} d x $$

Question

UpStudy AI · Accepted Answer

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.
Aquí están las soluciones a los ejercicios de integración: 1. $$ \int(3-x+x^{5}-2 x^{7}) d x = x^{2}+\frac{x^{6}}{6}-\frac{x^{8}}{4} + C $$ 2. $$ \int\left(5 \sqrt{x} \sqrt[3]{x}+\sqrt[4]{x}-\frac{2}{\sqrt[4]{x}}\right) d x = \frac{30}{11}x\sqrt[6]{x^{5}}+\frac{4}{5}x\sqrt[4]{x}-\frac{8}{3}\sqrt[4]{x^{3}} + C $$ 3. $$ \int(2-2 x)^{3} x d x = \frac{8-40x+80x^{2}-80x^{3}+40x^{4}-8x^{5}}{5}-2+8x-12x^{2}+8x^{3}-2x^{4} + C $$ 4. $$ \int \frac{3 x^{2}-4 x-4}{x-2} d x = \frac{3x^{2}}{2}+2x + C $$ 5. $$ \int \frac{(x+1)(x-2)}{\sqrt{x}} d x = \frac{2}{5}x^{2}\sqrt{x}-\frac{2}{3}x\sqrt{x}-4\sqrt{x} + C $$ Si necesitas más ayuda o aclaraciones sobre algún ejercicio, no dudes en preguntar.

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. \( \int \frac{(x+1)(x-2)}{\sqrt{x}} d x \)

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Calculus Questions