Bounds Integral of x times arcsine

In summary, the problem requires finding the bounds for the integral of x*arcsine(x) from 1/2 to 1. The approach suggested is to use the lower and upper bounds of arcsine(x) over the given interval, and then integrate the square of the lower and higher functions separately. This can be done using the closed form antiderivative of (arcsine(x))^2. However, this approach may be more difficult than necessary and Vela's approach of using the upper and lower bounds of arcsine(x) directly may be easier.
  • #1
FallenApple
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Homework Statement



Prove the integral of x*arcsine(x) from 1/2 to 1 is bounded between pi/16 and 3*pi/16

Homework Equations

The Attempt at a Solution


Not sure what to bound with. Do we use Squeeze Theorem?
 
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  • #2
The curves of ##y=x## and ##y=sin^{-1} x## start with the second below the first, then intersect so the second is above the first, then meet again at ##x=1##.

You can get a lower (upper) bound by integrating the square of the lower (higher) of the two from 1/2 up to the first intersection point, then doing the same again from that point up to 1 (noting that the lower and upper will have switched at the first intersection).

Wolfram tells me that ##(sin^{-1} x)^2## has a closed form antiderivative, so you should be able to obtain all derivatives.

That will give you upper and lower bounds, but I don't know if they are narrow enough to meet the problem spec. Worth a try anyway.

EDIT: Ignore this. Vela's approach is much easier. The above gives much tighter bounds but that is not required by the question. I was a little concerned about the degree of difficulty in this solution.
 
Last edited:
  • #3
FallenApple said:

Homework Statement



Prove the integral of x*arcsine(x) from 1/2 to 1 is bounded between pi/16 and 3*pi/16

Homework Equations

The Attempt at a Solution


Not sure what to bound with. Do we use Squeeze Theorem?
Over the interval given, what are the upper and lower bounds of ##\arcsin x##?
 
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What is the definition of bounds integral of x times arcsine?

The bounds integral of x times arcsine is a mathematical concept that involves finding the area under the curve of the function f(x) = x * arcsin(x) between two given bounds. This can be solved using calculus techniques such as integration.

What is the formula for calculating bounds integral of x times arcsine?

The formula for calculating the bounds integral of x times arcsine is ∫x * arcsin(x) dx = x * (arcsin(x) - 1) + √(1 - x^2) + C, where C is the constant of integration.

Why is the bounds integral of x times arcsine important in mathematics?

The bounds integral of x times arcsine is important in mathematics because it has applications in physics, engineering, and other fields where finding the area under a curve is necessary. It also helps in solving problems involving inverse trigonometric functions.

What are the properties of bounds integral of x times arcsine?

The properties of bounds integral of x times arcsine include linearity, where the integral of a sum of functions is equal to the sum of the integrals of each function, and the Fundamental Theorem of Calculus, which states that the integral of a function is equal to the difference of its values at the upper and lower bounds.

How do you solve a problem involving bounds integral of x times arcsine?

To solve a problem involving bounds integral of x times arcsine, you can follow these steps:
1. Identify the upper and lower bounds of the integral.
2. Use integration techniques to find the antiderivative of the function.
3. Substitute the upper and lower bounds into the antiderivative.
4. Subtract the value of the antiderivative at the lower bound from the upper bound to find the final answer.

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