What is the integral of $\frac{x}{2\sqrt{x+2}}$?

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  • Thread starter karush
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In summary, the conversation discusses using the integration by parts method to solve the integral $\int \frac{x}{2\sqrt{x+2}}\,dx$. The final answer is found to be $\frac{\left(x-4\right)\sqrt{x+3}}{3}$ and the discussion also includes a reminder to factor and simplify the expression.
  • #1
karush
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MHB
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$\tiny\text{LCC 206 {r9} IBP}$
$I=\int \frac{x}{2\sqrt{x+2}}\,dx
=\frac{\left(x-4\right)\sqrt{x+3}}{3}$
Rewrite as
$I=\int \frac{x}{2}\frac{1}{\sqrt{x+2}}\, dx $
Let
$\displaystyle
u=\frac{x}{2} \ \ \ dv=\frac{1}{\sqrt{x+2}} \, dx \\
du=\frac{1}{2} \ \ \ v=2\sqrt{x+2} \\ $
Then
$x\sqrt{x+2}-\int \sqrt{x+2} \ dx$
This doesn't look it's heading toward the answer😞

$\tiny\text
{from Surf the Nations math study group}$
🏄 🏄 🏄
 
Last edited:
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  • #2
You're doing fine...complete the last integral (with a constant of integration), then factor. :)
 
  • #3
$\tiny\text{LCC 206 {r9} IBP}$
$$\displaystyle
I=\int \frac{x}{2\sqrt{x+2}}\,dx
=\frac{\left(x-4\right)\sqrt{x+3}}{3} \\
\text{rewrite as} \\
I=\int \frac{x}{2}\frac{1}{\sqrt{x+2}}\, dx \\
u=\frac{x}{2} \ \ \ dv=\frac{1}{\sqrt{x+2}} \, dx \\
du=\frac{1}{2} \ \ \ v=2\sqrt{x+2} \\
\text{then} \\
I=x\sqrt{x+2}-\int \sqrt{x+2} \ dx \\
= x\sqrt{x+2}-\frac{2}{3}\left(x+2\right) \\
\text{factor } \\
=\sqrt{x+2}\left[x-\frac{2\left(x+2\right)}{3}\right] \\
\text{simplify} \\
I=\frac{\left(x-4\right)\sqrt{x+3}}{3}$$
$\tiny\text
{from Surf the Nations math study group}$
🏄 🏄 🏄 🏄🏄
 
Last edited:

FAQ: What is the integral of $\frac{x}{2\sqrt{x+2}}$?

1. What is LCC 206?

LCC 206 stands for "Library and Cultural Collections 206", a course offered at many universities that focuses on developing critical thinking and research skills through the examination of cultural artifacts and literature.

2. What is the purpose of LCC 206?

The purpose of LCC 206 is to provide students with the tools and knowledge to critically analyze and interpret various cultural artifacts, such as literature, art, and media. It also aims to develop research skills and promote cultural understanding and appreciation.

3. What is IBP in relation to LCC 206?

IBP stands for "Intercultural Business Practicum" and it is a component of LCC 206 that focuses on the application of intercultural skills in a business setting. This includes understanding cultural differences in communication, negotiation, and decision-making.

4. Is LCC 206 a required course for all students?

It depends on the university and the major. Some universities may require LCC 206 as a general education course, while others may offer it as an elective for certain majors. It is best to check with your academic advisor for specific requirements.

5. What can I expect to learn from LCC 206?

By taking LCC 206, you can expect to learn critical thinking skills, research methods, and cultural understanding and appreciation. You will also gain practical skills for applying intercultural knowledge in a business setting through the IBP component of the course.

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