int cos x^(2)e^(2x)dx

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Identify Integral: Identify the integral to be solved. We are given the integral int cos(x^2)e^(2x)dx, which we need to solve.Simplify Integral: Look for a substitution or a method to simplify the integral. This integral does not lend itself to simple methods of integration such as substitution or integration by parts, as the integrand is a product of a trigonometric function and an exponential function of different powers. We will attempt to use integration by parts.Apply Integration by Parts: Apply integration by parts. Integration by parts is given by the formula ∫u dv = uv - ∫v du, where u and dv are parts of the integrand. We need to choose u and dv such that the resulting integral is simpler.Choose u and dv: Choose u and dv. Let's choose u = cos(x^2) and dv = e^(2x)dx. Then we need to find du and v.Differentiate u: Differentiate u to find du. du = d/dx[cos(x^2)]dx = -2x sin(x^2)dxIntegrate dv: Integrate dv to find v. v = ∫e^(2x)dx = 1/2 e^(2x)Substitute into Formula: Substitute u, du, v, and dv into the integration by parts formula. ∫cos(x^2)e^(2x)dx = uv - ∫v du = cos(x^2)(1/2 e^(2x)) - ∫(1/2 e^(2x))(-2x sin(x^2))dxSimplify Expression: Simplify the expression. = 1/2 e^(2x)cos(x^2) + ∫x sin(x^2)e^(2x)dxAttempt to Solve: Attempt to solve the new integral ∫x sin(x^2)e^(2x)dx. This integral appears to be even more complex than the original one, and it is not clear how to proceed with elementary functions. This suggests that the integral of cos(x^2)e^(2x) with respect to x may not have a solution in terms of elementary functions.Conclude Solution: Conclude that the integral cannot be expressed in terms of elementary functions. The integral int cos(x^2)e^(2x)dx does not have a solution in terms of elementary functions. It may require special functions or numerical methods to evaluate.

$\int \cos(x^2)e^{2x}\,dx$

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∫cos⁡(x2)e2x dx\int \cos(x^2)e^{2x}\,dx∫cos(x2)e2xdx

$\int \cos(x^2)e^{2x}\,dx$