How Do You Integrate w/(w^4 + 1)?

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In summary, the conversation discusses different approaches to integrating the function w/w^4 + 1, including using substitution and integration by parts. It is suggested to substitute u = x^2 and then use the formula for integrating du/(u^2 + 1). The correct substitution is u = w^2, resulting in u^2 = w^4 and du = 2w dw.
  • #1
chubb rock
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I'm supposed to integrate w/w^4 + 1

The two ideas I'd come up with was trying to substitute w^4+1 with u but that results in a 0 in the numerator.

Then I thought maybe I have to put w/w^4+1 into the 1/1+w^2 form and integrate into 1/k tan^-1kx + C but I'm not sure how to keep the original function while making the w^4 a w^2.

Help please? Am I at least going in the right direction with any of these ideas?
 
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  • #2
Use integration by parts. You'll have to do it twice.

EDIT: Mark44's way is better, ignore this.
 
  • #3
You're going in the right direction thinking of tan-1(x).

An ordinary substitution should do the trick: u = x2, so du = 2xdx. Then your integrand becomes roughly du/(u2 + 1). Notice the parentheses I added to make clear what the denominator is. You'll also need to add the right constant multiplier in the numerator, since I omitted it.
 
  • #4
Does that mean I can substitute just the variable and leave the "+ 1"? That is substitute u as w^2 getting u^2 = u^4?
 
  • #5
You mean u2 = w4.
 
  • #6
Yes. The substitution is u = w2, so u2 = w4, and du = 2wdw.
 
  • #7
Oh, yeah. Typo. That's what I meant.

Awesome, thanks for the help.
 

Related to How Do You Integrate w/(w^4 + 1)?

1. What is the purpose of integrating w/w^4 + 1?

The purpose of integrating w/w^4 + 1 is to find the antiderivative of the given function. This can be useful in solving various problems in physics, engineering, and other scientific fields.

2. How do I integrate w/w^4 + 1?

The first step in integrating w/w^4 + 1 is to rewrite the function in a simpler form, such as w^-3 + 1. Then, you can use the power rule of integration to find the antiderivative. The final step is to add the constant of integration to the result.

3. What is the power rule of integration?

The power rule of integration states that the integral of x^n is equal to x^(n+1) / (n+1), where n is any real number except -1. In other words, to integrate a function, you can simply increase the power of x by 1 and divide by the new power.

4. Can I use a calculator to integrate w/w^4 + 1?

Yes, you can use a calculator to integrate w/w^4 + 1. Most scientific calculators have a built-in integration function that can handle basic integrals like this one. However, for more complex integrals, it is recommended to use software such as Mathematica or Wolfram Alpha.

5. What are some real-life applications of integrating w/w^4 + 1?

Integrating w/w^4 + 1 can be useful in solving problems related to motion, such as finding the displacement, velocity, or acceleration of an object. It can also be used in calculating work, energy, and other quantities in physics and engineering. In addition, integration is an important tool in statistics and probability, and can be used to find areas under curves, which have various applications in finance, economics, and other fields.

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