Solving cos^3 x dx: A Quick and Easy Method

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In summary, the solution for integrating cos^3 x dx is sinx - sin^3 x / 3 + C, which can be found by solving the integral of cosx*(1-sin^2x) dx. However, there may be confusion on how to solve cosx*(1-sin^x) dx and it may be helpful to mention that you are trying to integrate. One approach is to distribute the multiplication and see if that gives any hints. Another approach is to use substitution, which may be more familiar.
  • #1
PrudensOptimus
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OK, I know the solution for cos^3 x dx is sinx - sin^3 x / 3 + C.


And that

you basically solve

integral of cosx*(1-sin^2x) dx. to get it.

but,...

what I don't get is how do you solve cosx*(1-sin^x) dx... is there a trick that I didn't get from the parts formula?
 
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  • #2
It would help if you mentioned you're trying to integrate!

Distribute the multiplication and see if that gives you any hints.
 
  • #3
Yes, one doesn't normally say "solve f(x)dx"!

Hurkyl, I don't see any reason to "distribute" (multiply out) anything. There is an obvious substitution for ∫(1- sin2(x))cos(x)dx.
 
  • #4
Good point. :smile:
 
  • #5
I have never learned integration by parts. Please help me.
 
  • #6
But, I presume, you know substitution?
 
  • #7
Nope, any products in Integrals other than those constants are new to me.
 
  • #8
I didn't ask about products, I asked about substitution!


E.G. would you know how to integrate ∫ sin(πx) dx
 
  • #9
i know the answer,

but I don't know the part when they did the dx = dv(ax) part... that confuzed me.
 
  • #10
Ok well here's how I worked it out

http://myfiles.dyndns.org/pictures/integrate1.jpg

I put a few steps together but you can still see what happened sort of.
 
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1. What is the solution for cos^3 x dx?

The solution for cos^3 x dx is (sin x + C)/2, where C is the constant of integration.

2. How do you solve for cos^3 x dx?

To solve for cos^3 x dx, you can use the trigonometric identity cos^2 x = 1 - sin^2 x and then use integration by parts.

3. Why is the solution for cos^3 x dx (sin x + C)/2?

The solution for cos^3 x dx is (sin x + C)/2 because when you use integration by parts, the integral of cos^2 x is equal to (sin x)/2. Therefore, when you substitute in the identity cos^2 x = 1 - sin^2 x, you get the solution (sin x + C)/2.

4. Are there any other ways to solve for cos^3 x dx?

Yes, there are other methods for solving for cos^3 x dx such as using the substitution method or using the double angle formula for cos^2 x.

5. Can you explain why the solution for cos^3 x dx is an indefinite integral?

The solution for cos^3 x dx is an indefinite integral because it represents the antiderivative of the function cos^3 x. An indefinite integral does not have specific limits of integration and therefore, it represents the entire family of curves that have the same derivative (in this case, cos^3 x).

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