Integrate x^2/sqrt(4-x^2): Solution Steps

Thread starter futb0l
Start date Jul 15, 2006
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Integrate

In summary, to simplify the expression x^2/sqrt(4-x^2), we can rewrite it as x^2(4-x^2)^(-1/2) and then use the power rule to get x^2/((4-x^2)^(1/2)). The domain of this expression is all real numbers except for x=2 and x=-2. To integrate the expression, we can use substitution and the power rule to get -1/3 * (4-x^2)^(3/2) + C. This expression cannot be simplified further. It can be used in real-life applications to calculate areas and volumes, and in physics and engineering calculations involving circular motion and rotational energy.

Jul 15, 2006

futb0l

[tex]
\int{ \frac{x^2}{\sqrt{4-x^2}} }
[/tex]

How would I do that?
I tried integrating by parts, letting u = x^2 and v' = 1/sqrt(4-x^2) and many other combinations, but I just can't seem to get the result.

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Jul 15, 2006

matt grime

Science Advisor

Homework Helper

A trig substitution will work, surely.

Jul 15, 2006

HallsofIvy

Science Advisor

Homework Helper

Let [itex] x= 2 sin(\theta)[/itex]

Related to Integrate x^2/sqrt(4-x^2): Solution Steps

1. How do I simplify the expression x^2/sqrt(4-x^2)?

To simplify this expression, we can first rewrite it as x^2(4-x^2)^(-1/2). Then, we can use the power rule for exponents to bring down the exponent of -1/2 and rewrite the expression as x^2(4-x^2)^(-1/2) = x^2/((4-x^2)^(1/2)). This is the simplified form of the expression.

2. What is the domain of the expression x^2/sqrt(4-x^2)?

The domain of this expression is all real numbers except for x=2 and x=-2, since these values would result in division by zero which is undefined.

3. How do I integrate x^2/sqrt(4-x^2)?

To integrate this expression, we can first use the substitution u=4-x^2. Then, du=-2x dx, which we can use to rewrite the expression as -1/2 * (x^2/sqrt(u)) du. This is now in the form of a known integral, so we can use the power rule to integrate and get (-1/2) * (2/3) * u^(3/2) + C = -1/3 * (4-x^2)^(3/2) + C. Remember to substitute back in x for u to get the final answer.

4. Can this expression be simplified further?

No, this expression is already in its simplified form and cannot be further reduced.

5. How can I use this expression in real-life applications?

This expression can be used to calculate the area under a curve in certain scenarios, such as calculating the area of a circle or the volume of a sphere. It can also be used in physics and engineering calculations involving circular motion or rotational energy.