Anti-Derivative of cos(theta^2): Chain Rule?

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In summary, the anti-derivative of cos(theta^2) cannot be expressed in terms of elementary functions. To find the derivative, the chain rule must be used. Alternatively, it can be reduced to an infinite series and integrated term by term for a more accurate result. However, this may require more than 3 terms for great accuracy.
  • #1
afcwestwarrior
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whats the anti derivative of cos(theta^2)
do i use the chain rule
 
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  • #2
somebody could answer this and make me look stupid but I am going to guess this is a function whose antiderivative cannot be expressed as an elementary combination of the usual suspects.

if you think about it, using the chain rule would be expected to give you a product as an answer, so it is hard to imagine how to get this function as a derivative.

of course it is continuous, hence the ftc says it is the derivative of its indefinite integral.
 
  • #3
so i find the derivative of it
 
  • #4
hmmmm
 
  • #5
i thought it would be 2* sin (theta)^2
 
  • #6
the anti-derivative of [itex]cos(\theta^2)[/itex] can't be expressed in terms of elementary functions as mathwonk said.

But if you want to find the derivative of it, you'll need the chain rule.
 
  • #7
how would i find the antiderivative of it
 
  • #8
Not all functions have anti-derivatives that can expressed in familiar form (ie. elementary form) This would be one of those.
 
  • #9
afcwestwarrior said:
how would i find the antiderivative of it

Look up the MacLaurin of [tex]cos(\theta)[/tex], from there figure out what the MacLaurin would be for [tex]cos(\theta^{2}) [/tex]. Now you can integrate it out and see what you get.
 
  • #10
afcwestwarrior said:
how would i find the antiderivative of it

Whenever a function is difficult or even impossible to integrate in terms of the elementary functions, you can always reduce it to an infinite series and integrate it term by term.

The series for cos (x) is 1 – x^2/2! + x^4/4! – x^6/6! . . . . .

Just plug x^2 in there in place of x and get:

1 – x^4/2! + x^8/4! – x^12/6! . . . . .

Now just integrate that term by term. Because of the factorials in the denominator the series converges quickly so only 3 terms should be needed unless you require great accuracy.
 
  • #11
ok i get it, I'm very slow like a turtle,

"The mind may be slow at times, but through time the information will be gathered."
Eugeno Ponce
 

FAQ: Anti-Derivative of cos(theta^2): Chain Rule?

What is the chain rule?

The chain rule is a rule in calculus that is used to find the derivative of a composite function. It states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.

How do you apply the chain rule to find the anti-derivative of cos(theta^2)?

To apply the chain rule to find the anti-derivative of cos(theta^2), you must first identify the outer function and the inner function. In this case, the outer function is cos(x) and the inner function is theta^2. Then, using the chain rule formula, you can find the anti-derivative by taking the anti-derivative of the outer function and multiplying it by the derivative of the inner function.

What is the anti-derivative of cos(theta^2)?

The anti-derivative of cos(theta^2) is equal to sin(theta^2) times the derivative of theta^2, which is 2theta. Therefore, the anti-derivative of cos(theta^2) is 2theta sin(theta^2).

Can the chain rule be used to find the anti-derivative of any function?

Yes, the chain rule can be used to find the anti-derivative of any function that is a composite function. However, the specific steps may vary depending on the function and its composition.

How do you check your answer when using the chain rule to find the anti-derivative?

To check your answer, you can take the derivative of the anti-derivative you found using the chain rule. If the result is equal to the original function, then your answer is correct. You can also use online derivative calculators to verify your answer.

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