Integration of (x)/[(4-x^2)^0.5]

In summary, the integration of (x)/[(4-x^2)^0.5] is equal to arcsin(x/2) + C, and can be simplified to 2arcsin(x/2) + C using the trigonometric identity sin(2θ) = 2sin(θ)cos(θ). The domain of the integrand is all real numbers except for -2 and 2, and the range is between -π/2 and π/2. It can be used in real-world applications to calculate displacement, velocity, and acceleration, and represents the area under a semicircle with a radius of 2.
  • #1
ZedCar
354
1

Homework Statement


integrate

(x)/[(4-x^2)^0.5]



Homework Equations





The Attempt at a Solution



which method would be used to integrate this, or is it just a standard integral from a regular standard integral table, so can be integrated directly? The book seems to indicate it is a standard integral, but I can't seem to locate it.

I can see a similar standard integral which is

(1)/[a^2 - x^2]^0.5

though this is 1 over, and the question is x over.


Thank you
 
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  • #2
A simple u-substitution would work quite well here.
 

FAQ: Integration of (x)/[(4-x^2)^0.5]

What is the integration of (x)/[(4-x^2)^0.5]?

The integration of (x)/[(4-x^2)^0.5] is equal to arcsin(x/2) + C. This can be derived using the substitution method and the trigonometric identity sin^2(x) + cos^2(x) = 1.

Can the integration of (x)/[(4-x^2)^0.5] be simplified?

Yes, the integration can be simplified by using the trigonometric identity sin(2θ) = 2sin(θ)cos(θ). This will result in the simplified form of 2arcsin(x/2) + C.

What is the domain and range of the integrand (x)/[(4-x^2)^0.5]?

The domain of the integrand is all real numbers except for -2 and 2, as these values will result in division by zero. The range is all real numbers between -π/2 and π/2, as this is the range of the arcsine function.

Can the integration of (x)/[(4-x^2)^0.5] be used in real-world applications?

Yes, the integration of (x)/[(4-x^2)^0.5] can be used in physics and engineering to calculate the displacement, velocity, and acceleration of an object moving in a circular path.

What is the geometric interpretation of the integration of (x)/[(4-x^2)^0.5]?

The integration of (x)/[(4-x^2)^0.5] represents the area under the curve of a semicircle with a radius of 2. This can be visualized by graphing the integrand and observing the shape of the curve.

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