Final answer:
The integral of cos(x²) is evaluated using special functions, specifically the Fresnel integrals, rather than elementary methods like trigonometric substitution, U-substitution, or integration by parts. In trigonometry, important identities like sin(2θ) = 2sinθcosθ and cos(2θ) = cos²θ - sin²θ are used in applications such as wave analysis.
Step-by-step explanation:
The integral of cos(x²) cannot be expressed in terms of elementary functions and therefore does not have an elementary antiderivative. Thus, options such as trigonometric substitution, U-substitution, and integration by parts will not result in a solvable integral for cos(x²). Instead, the integral of cos(x²) is commonly evaluated using special functions, specifically the Fresnel integrals, which are used in the fields of wave optics and quantum mechanics.
To evaluate a line integral as seen in the given reference material, one would typically reduce it to an integral over a single variable or parameter. The process often involves choosing the parameter that simplifies the integral the most, avoiding more complex functions such as those involving square roots or fractional exponents where possible.
In relation to trigonometry, there are several identities provided in the reference material such as sin(2θ) = 2sinθcosθ and cos(2θ) = cos²θ - sin²θ, which are crucial for various applications including the analysis of waves, as described in the challenge problem concerning the phase shift of wave functions.