Solve Integral of cos(y^2) - Calculus

In summary, the conversation is about finding the integral of cos(y^2) and whether it can be simplified to terms of sin(y) and cos(y). One person suggests using the identity cos(2y)=cos(y+y) to simplify it, but another points out that it is actually cos(y^2) not cos(2y). The final response states that there is no elementary solution, but the anti derivative can be expressed in terms of the Cosine Fresnel Integral.
  • #1
caaron3
2
0
Can anyone help me with the integral of cos(y^2)?
 
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  • #2
Possibly. Can you reduce cos(y^2) down and express it in terms of sin(y) and cos(y)?
 
  • #3
not really. I'm finding functions, given the gradient...

((e^x)*cos(y^2))i - (2y(e^x)sin(y^2))j
 
  • #4
I reckon you can. How about if you write cos(2y)=cos(y+y)=...
 
  • #5
cristo - its cos (y^2), not 2y :(

caaron3 - it has no elementary solution, though if you really want some sort of a solution, the anti derivative is in terms of the Cosine Fresnel Integral, so look that up.
 

FAQ: Solve Integral of cos(y^2) - Calculus

What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is used in calculus to find the total value of a continuously changing quantity.

How do you solve an integral?

To solve an integral, you must first identify the function that needs to be integrated. Then, you can use various techniques such as substitution, integration by parts, or trigonometric identities to simplify the integral and find its solution.

What is a definite integral?

A definite integral is an integral with specific limits of integration. It represents the area under a curve between two specified points on a graph.

How do you solve the integral of cos(y^2)?

To solve the integral of cos(y^2), you can use the substitution method by letting u = y^2. This will transform the integral into ∫cos(u) du, which can be solved using the trigonometric identity cos(u) = (1/2)(1 + cos(2u)).

Can you explain the steps for solving an integral using the substitution method?

Yes, the steps for solving an integral using the substitution method are as follows:

  1. Identify the function that needs to be integrated.
  2. Choose an appropriate substitution, usually by letting u = some expression within the function.
  3. Calculate the derivative of u, du, and substitute it into the integral, replacing all instances of the original variable with u and all instances of its derivative with du.
  4. Simplify the integral using algebra and trigonometric identities.
  5. Integrate the simplified integral and substitute back in the original variable to get the final solution.

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