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

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Gaussian integral

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Q.

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

Identify Integral: Identify the integral to be solved. $\newline$ 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 $\int u \, dv = uv - \int v \, du$ , 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 $du$ . $du = \frac{d}{dx}[\cos(x^2)]dx = -2x \sin(x^2)dx$
Integrate dv: Integrate dv to find v. $\newline$ v = $\int e^{(2x)}dx = \frac{1}{2} e^{(2x)}$
Substitute into Formula: Substitute $u$ , $du$ , $v$ , and $dv$ into the integration by parts formula. $\newline$ $\int \cos(x^2)e^{2x}dx = uv - \int v du$ $\newline$ = $\cos(x^2)(\frac{1}{2} e^{2x}) - \int(\frac{1}{2} e^{2x})(-2x \sin(x^2))dx$
Simplify Expression: Simplify the expression. $\newline$ $= \frac{1}{2} e^{(2x)}\cos(x^2) + \int x \sin(x^2)e^{(2x)}dx$
Attempt to Solve: Attempt to solve the new integral $\int x \sin(x^2)e^{(2x)}\,dx$ . 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. $\newline$ 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.

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