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

Problem

(∫_^)(cos(x2)*d(x))

Solution

1. Identify the integral as a non-elementary integral that cannot be expressed in terms of standard algebraic or trigonometric functions.

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

3. Apply the power series expansion for cos(u) where u=x2 to express the integral as an infinite series.

4. Substitute the series cos(x2)=(∑_n=0^∞)(((−1)n*(x2)(2*n))/((2*n)!)) into the integral.

5. Simplify the expression to (∑_n=0^∞)(((−1)n*x(4*n))/((2*n)!))

6. Integrate term by term with respect to x

7. Add the constant of integration C

Final Answer

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

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