Evaluate the Integral integral of sin(x^2) with respect to x

Problem

(∫_^)(sin(x2)*d(x))

Solution

1. Identify the integral as a non-elementary integral, meaning it cannot be expressed in terms of basic functions like polynomials, logarithms, or standard trigonometric functions.

2. Recognize the form of the integral as a Fresnel integral, specifically the Fresnel S integral, which is defined as S(x)=(∫_0^x)(sin(t2)*d(t))

3. Apply the power series expansion for sin(u) where u=x2 to find an infinite series representation.

4. Substitute u=x2 into the Taylor series sin(u)=(∑_n=0^∞)(((−1)n*u(2*n+1))/((2*n+1)!))

5. Integrate the resulting series term by term: (∫_^)((∑_n=0^∞)(((−1)n*x(4*n+2))/((2*n+1)!))*d(x))

6. Simplify the result to obtain the general antiderivative in terms of the Fresnel S function or the infinite series.

Final Answer

(∫_^)(sin(x2)*d(x))=(∑_n=0^∞)(((−1)n*x(4*n+3))/((4*n+3)*(2*n+1)!))+C

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