Integrating -2x Over (1-x^2)^(1/2): A Confusing Challenge

In summary, the antiderivative of -2x/(1-x^2)^(1/2) can be found by making a substitution, u = 1-x^2, and using the formula for du = -2x dx. This simplifies the integral to a form that is easier to solve. It is important to analyze the problem and try different substitutions to find the one that works best.
  • #1
Jacobpm64
239
0
Find the antiderivative of:

-2x
(1-x^2)^(1/2)

That's -2x over all of that... Bleh, i suck at code.

But anyway.. I just started integrals, and this is confusing for me... Is there a product or quotient rule in integrals like there is in derivatives?.. if not? how do you work this?
 
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  • #2
Make a substitution, let u = 1-x2 then du = -2x and you have a fairly simple integral in terms of u.
 
  • #3
Here is the code for it in case you need to use it in the future (click to see code)

[tex]\int\frac{-2x}{\sqrt{1-x^2}} \ dx[/tex]
 
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  • #4
how do you know what to make u and what to make du?
 
  • #5
By thinking and analyzing the problem! The difficulty is that [itex]\sqrt{1- x^2}[/itex] in the denominator. The very first thing you should have thought about was substituting for that: u= 1- x^2. Then you would immediately see that du= -2xdx and that would work only if you had an xdx already in the problem (the "-2" is a constant and you can move constants in and out of the integral- you can't do that with "x", the x has to already be there) and, indeed, the problem has an "xdx". Lucky you!

(Once you've decided what to try for u, you don't "decide" what du is- that follows from the derivative of u. Notice I said "try"- you can seldom look at an integral and know what substitution will work. A lot of it is trial and error.)
 

1. What is the purpose of integrating -2x over (1-x^2)^(1/2)?

The purpose of integrating -2x over (1-x^2)^(1/2) is to find the area under the curve of the given function. This process is known as integration and is a fundamental concept in calculus.

2. Why is integrating -2x over (1-x^2)^(1/2) considered a confusing challenge?

The integration of -2x over (1-x^2)^(1/2) can be challenging because it involves using a combination of substitution, trigonometric identities, and integration by parts. This can be overwhelming for students who are not familiar with these techniques.

3. What are the steps involved in integrating -2x over (1-x^2)^(1/2)?

Step 1: Use substitution to simplify the integrand.Step 2: Apply trigonometric identities to transform the integrand into a more manageable form.Step 3: Use integration by parts to integrate the transformed function.Step 4: Simplify the resulting expression and solve for the definite or indefinite integral.

4. Are there any tips for solving the integration of -2x over (1-x^2)^(1/2)?

Some tips for solving the integration of -2x over (1-x^2)^(1/2) include being familiar with the properties of trigonometric functions, practicing substitution techniques, and breaking the problem down into smaller, more manageable steps.

5. How is integrating -2x over (1-x^2)^(1/2) used in real-life applications?

The integration of -2x over (1-x^2)^(1/2) is used in various fields such as physics, engineering, and economics. It can be used to calculate the work done by a variable force, the velocity of an object under the influence of a variable acceleration, or the change in value of a stock over time.

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