1. The problem is to find the integral with respect to $x$ of the function $5e^x - 2e^{4x^2}$. 2. Recall the integral rules: - The integral of $e^{ax}$ with respect to $x$ is $\frac{1}{a}e^{ax} + C$. - For functions like $e^{4x^2}$, substitution is needed since the exponent is not linear. 3. Start with the integral: $$\int (5e^x - 2e^{4x^2}) \, dx = \int 5e^x \, dx - \int 2e^{4x^2} \, dx$$ 4. Integrate the first term: $$\int 5e^x \, dx = 5 \int e^x \, dx = 5e^x + C_1$$ 5. For the second term $\int 2e^{4x^2} \, dx$, substitution is not straightforward because $4x^2$ is quadratic and does not have an elementary antiderivative in terms of elementary functions. 6. Therefore, the integral of $e^{4x^2}$ cannot be expressed in elementary functions. We leave it as an integral or express it in terms of the error function if needed. 7. Final answer: $$5e^x - 2 \int e^{4x^2} \, dx + C$$ This is the most simplified form for the integral of the given function.