Find the Integral cos(x^2)

Problem

(∫_^)(cos(x2)*d(x))

Solution

1. Identify the integral as a non-elementary integral. The function cos(x2) does not have an antiderivative that can be expressed in terms of basic algebraic, trigonometric, or exponential functions.

2. Recognize the form of the Fresnel Integral. The Fresnel integral C(x) is defined specifically to represent the antiderivative of cos(t2) with a normalization factor.

3. Apply the definition of the Fresnel C function. By standard convention, the integral is expressed using the Fresnel C function, which is defined as C(x)=(∫_0^x)(cos((π*t2)/2)*d(t))

4. Adjust for the argument x2 To match the standard form C(x) a substitution or scaling is used to account for the π/2 factor in the standard definition.

5. State the result in terms of the Fresnel C function.

Final Answer

(∫_^)(cos(x2)*d(x))=√(,π/2)*C*(x√(,2/π))+C

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