12.1 Determine the following integrals:
(a)
©

Ask by Byrd Carrillo. in South Africa

Feb 02,2025

Question

12.1 Determine the following integrals: (a) $$ \int\left(\sqrt[3]{x}+\frac{1}{x}\right)^{2} d x $$ (c) $$ \int \frac{5 x}{\sqrt{x-2}} d x $$

UpStudy AI · Accepted Answer

Calculate the integral $$ \int(\sqrt[3]{x}+\frac{1}{x})^{2} dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \left(\sqrt[3]{x}+\frac{1}{x}ight)^{2} dx$$
- step1: Evaluate the power:
  $$\int \left(x^{\frac{1}{3}}+\frac{1}{x}ight)^{2} dx$$
- step2: Calculate:
  $$\int \left(x^{\frac{2}{3}}+\frac{1}{x^{\frac{2}{3}}}+\frac{1}{x^{\frac{2}{3}}}+\frac{1}{x^{2}}ight) dx$$
- step3: Use properties of integrals:
  $$\int x^{\frac{2}{3}} dx+\int \frac{1}{x^{\frac{2}{3}}} dx+\int \frac{1}{x^{\frac{2}{3}}} dx+\int \frac{1}{x^{2}} dx$$
- step4: Evaluate the integral:
  $$\frac{x^{\frac{5}{3}}}{\frac{5}{3}}+\int \frac{1}{x^{\frac{2}{3}}} dx+\int \frac{1}{x^{\frac{2}{3}}} dx+\int \frac{1}{x^{2}} dx$$
- step5: Evaluate the integral:
  $$\frac{x^{\frac{5}{3}}}{\frac{5}{3}}+\frac{x^{\frac{1}{3}}}{\frac{1}{3}}+\int \frac{1}{x^{\frac{2}{3}}} dx+\int \frac{1}{x^{2}} dx$$
- step6: Evaluate the integral:
  $$\frac{x^{\frac{5}{3}}}{\frac{5}{3}}+\frac{x^{\frac{1}{3}}}{\frac{1}{3}}+\frac{x^{\frac{1}{3}}}{\frac{1}{3}}+\int \frac{1}{x^{2}} dx$$
- step7: Evaluate the integral:
  $$\frac{x^{\frac{5}{3}}}{\frac{5}{3}}+\frac{x^{\frac{1}{3}}}{\frac{1}{3}}+\frac{x^{\frac{1}{3}}}{\frac{1}{3}}+\frac{x^{-1}}{-1}$$
- step8: Rewrite the expression:
  $$\frac{3x^{\frac{5}{3}}}{5}+\frac{x^{\frac{1}{3}}}{\frac{1}{3}}+\frac{x^{\frac{1}{3}}}{\frac{1}{3}}+\frac{x^{-1}}{-1}$$
- step9: Rewrite the expression:
  $$\frac{3x^{\frac{5}{3}}}{5}+3x^{\frac{1}{3}}+\frac{x^{\frac{1}{3}}}{\frac{1}{3}}+\frac{x^{-1}}{-1}$$
- step10: Rewrite the expression:
  $$\frac{3x^{\frac{5}{3}}}{5}+3x^{\frac{1}{3}}+3x^{\frac{1}{3}}+\frac{x^{-1}}{-1}$$
- step11: Rewrite the expression:
  $$\frac{3x^{\frac{5}{3}}}{5}+3x^{\frac{1}{3}}+3x^{\frac{1}{3}}-\frac{1}{x}$$
- step12: Add the terms:
  $$\frac{3x^{\frac{5}{3}}}{5}+6x^{\frac{1}{3}}-\frac{1}{x}$$
- step13: Simplify:
  $$\frac{3x\sqrt[3]{x^{2}}}{5}+6x^{\frac{1}{3}}-\frac{1}{x}$$
- step14: Simplify:
  $$\frac{3x\sqrt[3]{x^{2}}}{5}+6\sqrt[3]{x}-\frac{1}{x}$$
- step15: Add the constant of integral C:
  $$\frac{3x\sqrt[3]{x^{2}}}{5}+6\sqrt[3]{x}-\frac{1}{x} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{5x}{\sqrt{x-2}} dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using partial integration formula:
  $$\int \frac{5x}{\sqrt{x-2}} dx$$
- step1: Rewrite the expression:
  $$\int 5	imes \frac{x}{\sqrt{x-2}} dx$$
- step2: Use properties of integrals:
  $$5	imes \int \frac{x}{\sqrt{x-2}} dx$$
- step3: Evaluate the integral:
  $$5	imes \int \left(\sqrt{x-2}ight)^{-1}x dx$$
- step4: Prepare for integration by parts:
  $$\begin{align}&u=x\&dv=\frac{1}{\sqrt{x-2}}dx\end{align}$$
- step5: Calculate the derivative:
  $$\begin{align}&du=dx\&dv=\frac{1}{\sqrt{x-2}}dx\end{align}$$
- step6: Evaluate the integral:
  $$\begin{align}&du=dx\&v=2\left(x-2ight)^{\frac{1}{2}}\end{align}$$
- step7: Substitute the values into formula:
  $$5\left(x	imes 2\left(x-2ight)^{\frac{1}{2}}-\int 1	imes 2\left(x-2ight)^{\frac{1}{2}} dxight)$$
- step8: Calculate:
  $$5\left(2x\left(x-2ight)^{\frac{1}{2}}-\int 2\left(x-2ight)^{\frac{1}{2}} dxight)$$
- step9: Calculate:
  $$10x\left(x-2ight)^{\frac{1}{2}}-5	imes \int 2\left(x-2ight)^{\frac{1}{2}} dx$$
- step10: Evaluate the integral:
  $$10x\left(x-2ight)^{\frac{1}{2}}-\frac{20\left(x-2ight)^{\frac{3}{2}}}{3}$$
- step11: Simplify:
  $$10x\sqrt{x-2}-\frac{20\left(x-2ight)^{\frac{3}{2}}}{3}$$
- step12: Simplify:
  $$10x\sqrt{x-2}-\frac{20\sqrt{\left(x-2ight)^{3}}}{3}$$
- step13: Expand the expression:
  $$10x\sqrt{x-2}-\frac{\left(20x-40ight)\sqrt{x-2}}{3}$$
- step14: Add the constant of integral C:
  $$10x\sqrt{x-2}-\frac{\left(20x-40ight)\sqrt{x-2}}{3} + C, C \in \mathbb{R}$$
The integral of $$ \int\left(\sqrt[3]{x}+\frac{1}{x}ight)^{2} dx $$ is $$ \frac{3x\sqrt[3]{x^{2}}}{5}+6\sqrt[3]{x}-\frac{1}{x} + C $$, where $$ C $$ is a constant.

The integral of $$ \int \frac{5x}{\sqrt{x-2}} dx $$ is $$ 10x\sqrt{x-2}-\frac{(20x-40)\sqrt{x-2}}{3} + C $$, where $$ C $$ is a constant.
The integrals are: 1. $$ \int\left(\sqrt[3]{x}+\frac{1}{x}ight)^{2} dx = \frac{3x\sqrt[3]{x^{2}}}{5}+6\sqrt[3]{x}-\frac{1}{x} + C $$ 2. $$ \int \frac{5x}{\sqrt{x-2}} dx = 10x\sqrt{x-2}-\frac{(20x-40)\sqrt{x-2}}{3} + C $$ where $$ C $$ is a constant.

12.1 Determine the following integrals:
(a)
©

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Calculus Questions

12.1 Determine the following integrals: (a) ©