# What is the integral of #cos(t^2)#?

graph{(y - cos(x^2))(y - 1 + x^4/2 - x^8/24 + x^12/720 - x^(10)/(40320)) = 0 [-4.934, 4.935, -2.464, 2.47]}

Here, the bottom graph is the estimate graph from the Maclaurin series.

To compare, I used Wolfram Alpha to get:

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There is no elementary solution.

We want to find:

As indicated in the alternative answers we can find a Maclaurin Series expansion, Unfortunately there is no elementary solution to the integral.

Instead, Numerical techniques are used to evaluate the following definite integrals:

Which are known as the Fresnel integrals.

Typically the values of these function are calculated using computer algorithms, or looked up in tables (in a similar way to that of the Normal Distribution ).

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graph{(y - cos(x^2))(y - 1 + x^4/2 - x^8/24 + x^12/720 - x^(10)/(40320)) = 0 [-4.934, 4.935, -2.464, 2.47]}

Here, the bottom graph is the estimate graph from the Maclaurin series.

To compare, I used Wolfram Alpha to get:

By signing up, you agree to our Terms of Service and Privacy Policy

The integral of ( \cos(t^2) ) does not have a standard antiderivative in terms of elementary functions. It cannot be expressed using standard functions like polynomials, exponentials, logarithms, trigonometric functions, and their inverses. Therefore, its integral is considered non-elementary and is typically expressed using special functions such as the Fresnel integral or the Dawson function.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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