Find the Integral sin(x^2)

Problem

(∫_^)(sin(x2)*d(x))

Solution

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

2. Recognize the standard form of the Fresnel Integral. The Fresnel Sine Integral, denoted as S(x) is defined by a specific power series or integral form.

3. Apply the standard definition of the Fresnel Sine Integral. By convention, the integral is often expressed using the function S(x)

4. Express the result using the power series expansion if a closed form is not possible. The power series for sin(x2) is (∑_n=0^∞)(((−1)n*x(4*n+2))/((2*n+1)!))

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

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

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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