How do you intergrate cos^3 [x] .dx

  • Thread starter suddy72
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In summary, the conversation discusses the steps for solving the integral problem of \int cos^3(x)dx. The key is to take out one of the cos(x) terms to match with the dx and then use the trig identity sin^2(x)+ cos^2(x)= 1 to convert to the other trig function. The conversation also mentions the use of LaTex Math Typesetting for writing mathematical symbols in messages and provides a website for checking integral problems.
  • #1
suddy72
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could some one please show me the steps to doing this ??
 
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  • #2
[itex] \int cos^3(x)dx [/itex] is a standard problem. Anytime you have an odd power of either sin or cos take one of the out to match with the dx:

[tex]\int(cos^2(x))(cos(x)dx)[/tex]

Now we can use [itex]sin^2(x)+ cos^2(x)= 1[/itex] to convert to the other trig function:

[tex]\int(1- sin^2(x))(cos(x)dx)[/tex]

and, of course, the whole reason for taking out that orginal "cos(x)" is that
d(sin(x))= cos(x)dx.

Let u= sin(x). Then du= cos(x)dx and the integral is

[tex]\int(1-u^2)du[/tex]
and that should be easy.
 
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  • #3
Well, I beg your forgiveness for this question b/c its really an irrelevant one.

Can you please tell me how you use these mathematical symbols like integral sign and superscripts etc in your messages?
 
  • #4
Xishan said:
Well, I beg your forgiveness for this question b/c its really an irrelevant one.

Can you please tell me how you use these mathematical symbols like integral sign and superscripts etc in your messages?
It's called LaTex Math Typesetting and is really pretty cool. Check out this thread for more information:
https://www.physicsforums.com/showthread.php?t=8997
 
  • #5
suddy, if you want a place to check your work, the integrator works fine--- powered by mathematica.

http://integrals.com/
 

1. What is the general process for integrating cos^3[x] .dx?

The general process for integrating cos^3[x] .dx is to use the power reduction formula to convert cos^3[x] into a product of cos[x] and cos^2[x]. Then, use the substitution method to rewrite cos^2[x] in terms of sin[x]. Finally, integrate the resulting expression using the power rule.

2. How do you use the power reduction formula to integrate cos^3[x] .dx?

To use the power reduction formula, you need to rewrite cos^3[x] as cos[x] * cos^2[x]. Then, use the identity cos^2[x] = 1/2 * (1 + cos[2x]) to expand the expression. This will give you an expression with both cos[x] and sin[x], which can then be integrated using the substitution method.

3. Can you explain the substitution method for integrating cos^3[x] .dx?

The substitution method involves substituting a variable, usually u, for a part of the original expression. In this case, we would substitute u = sin[x], which will allow us to rewrite cos^3[x] as (1 - u^2) * u. This can then be integrated using the power rule.

4. Do you need to use trigonometric identities when integrating cos^3[x] .dx?

Yes, in order to integrate cos^3[x] .dx, you will need to use the power reduction formula and possibly other trigonometric identities to simplify the expression and make it easier to integrate. Without using these identities, the integration process would be much more complicated.

5. Are there any special cases when integrating cos^3[x] .dx?

One special case is when the exponent of cos[x] is an odd number. In this case, the power reduction formula cannot be directly applied. Instead, you can use the identity cos^n[x] = cos[x] * cos^(n-1)[x] to rewrite the expression as a product of cos[x] and an even power of cos[x], which can then be integrated using the power reduction formula.

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