Find the Antiderivative sin(x^2)

Problem

(∫_^)(sin(x^2)*d(x))

Solution

1. Identify the integral as a non-elementary integral. The function sin(x^2) does not have an antiderivative that can be expressed in terms of basic algebraic, trigonometric, or exponential functions.

2. Recognize the standard form of the Fresnel Integral. The Fresnel Sine Integral is defined as S(x)=(∫_0^x)(sin(t^2)*d(t))

3. Apply the power series expansion for sin(u) where u=x^2 to express the antiderivative as an infinite series.

4. Substitute x^2 into the Taylor series for sine: sin(x^2)=(∑_n=0^∞)(((−1)^n*x^(4*n+2))/((2*n+1)!))

5. Integrate the series term by term to find the general antiderivative.

6. Express the result using the Fresnel S function notation, which is the conventional way to represent this specific antiderivative.

Final Answer

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

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