Calculus problem: Integral of sqrt(1/4-(x+3)^2

In summary, the individual is trying to solve an integral equation involving the square root of 1/4-(x+3)^2 but is having difficulty following the directions and getting lost. After simplifying the equation, the individual figured out that sin2A+cos2A=1. They also asked for help and showed their work in a file.
  • #1
ckwan48
4
0
I've been trying to solve this integral for a very long time!
Problem: Integral of sqrt(1/4-(x+3)^2

Effort: I have tried multiplying by sqrt(1/4) to get rid of the 4 so it can become a 1, so it will match up with inverse sine's derivative. After that I got kind of lost, since I don't know what happens or what will (x+3)^2, will become. I know what the answer should be like, but I want to do it without using that special formula, that they provide.

I know there's some trick because I didn't in high school, however, I don't recall it. I don't want to use that simple formula because it doesn't really tell you what's happening in the integration process. Can someone help me out? Thanks!
 
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  • #2
[tex] \int \sqrt{\frac{1}{4-(x+3)^2}}dx [/tex]

If that is your integral, you could simplify this to

[tex] \int \frac{1}{\sqrt{4-(x+3)^2}}dx[/tex]

Then try a trig substitution.

Think of sin2A+cos2A=1
 
  • #3
Yeah that is, I know it's suppose to be inverse sine and then I try getting rid of the 4 by multiplying sqrt(1/4) and then I get lost. Extra hints would help, thanks!
 
  • #4
ckwan48 said:
Yeah that is, I know it's suppose to be inverse sine and then I try getting rid of the 4 by multiplying sqrt(1/4) and then I get lost. Extra hints would help, thanks!

Can you show your steps? It might be hard to give you a hint when I don't really know where you get lost.
 
  • #5
I've attached a file, of what I have done so far. The "question mark" is where I'm stuck. And am I doing this step, right? Thanks!
 

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  • #6
It might help to do the substitution u = (x+3)/2
 
  • #7
i figured it out! i used u-substitution to make the integral simpler and then i factored out the 4.
 

1. What is the purpose of finding the integral of this calculus problem?

The integral of this calculus problem allows us to find the area underneath the curve of the given function. This can be useful in various applications, such as determining the displacement or velocity of an object.

2. How do I approach solving this integral?

To solve this integral, you can use the substitution method by substituting u = (x+3). This will change the integral to ∫√(1/4-u^2)du. Then, you can use the trigonometric substitution method by letting u = (1/2)sinθ, which will give you the integral of ∫cosθ dθ. From there, you can use trigonometric identities to solve for the final answer.

3. What is the domain of this function?

The given function has a square root, which means the radicand (1/4-(x+3)^2) must be greater than or equal to 0. This gives us the domain of x ≤ -3. However, since we are finding the integral of the function, we can extend the domain to include all real numbers.

4. Can I use a calculator to solve this integral?

Yes, you can use a calculator to solve this integral. However, it is important to understand the steps and concepts involved in solving the integral by hand before relying on a calculator.

5. What are some real-world applications of this calculus problem?

This calculus problem can be applied in physics, engineering, and other fields to determine the area under a curve and the displacement or velocity of an object. It can also be used in economics and business to calculate profits or losses by finding the area under a demand or supply curve.

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