<p>Let \(\displaystyle\int e^{x^2} \cdot e^x(2x^2 + x + 1)\,dx = e^{x^2} \cdot f(x) + C\) where \(f(x)\) is some non-zero constant function and \(C\) is some arbitrary constant. If the local minimum value of \(f(x)\) is equal to \(m\), then:</p>
<p>\(f(x)\) is increasing in \((0, \infty)\)</p>
<p>the value of \(\displaystyle\lim_{x \to 0}(1 + f(x))^{1/x}\) is equal to 1.</p>
<p>the value of \(\displaystyle\int_0^1 (f(x) + e^x)\,dx\) is equal to \(2e\).</p>
<p>the value of \(\left[\dfrac{-1}{m}\right]\) is equal to 2. [Note: \([\cdot]\) denotes greatest integer function.]</p>
Step-by-Step Solution
<div class="solution">
<p><strong>Step 1:</strong> To find \(f(x)\), we start with the given integral equation \(\displaystyle\int e^{x^2} \cdot e^x(2x^2 + x + 1)\,dx = e^{x^2} \cdot f(x) + C\). Let's simplify the integral using substitution. Notice that \(e^{x^2} \cdot e^x = e^{x^2 + x}\), so we are looking for a substitution that simplifies \(e^{x^2 + x}(2x^2 + x + 1)\).</p>
<p><strong>Step 2:</strong> Consider the derivative of \(x^2 + x\) which is \(2x + 1\). This is close to the \(2x^2 + x + 1\) term in our integral, except for the \(2x^2\) part. However, noticing that the derivative of \(x^2 + x\) does not directly help with \(2x^2 + x + 1\), we should look for a method to directly integrate \(e^{x^2} \cdot e^x(2x^2 + x + 1)\). Since \(e^{x^2} \cdot e^x = e^{x^2 + x}\), let's consider the structure of the integral and how it relates to \(f(x)\). The integral can be seen as \(\displaystyle\int e^{x^2 + x}(2x^2 + x + 1)\,dx\), suggesting a potential substitution or integration by parts could simplify this expression.</p>
<p><strong>Step 3:</strong> To proceed, let's examine the relationship between the integrand and \(f(x)\) more closely. Given \(\displaystyle\int e^{x^2} \cdot e^x(2x^2 + x + 1)\,dx = e^{x^2} \cdot f(x) + C\), differentiating both sides with respect to \(x\) yields \(e^{x^2} \cdot e^x(2x^2 + x + 1) = e^{x^2} \cdot (f(x) + f'(x) \cdot x + 2xf(x))\), simplifying the equation to find \(f(x)\). However, this step was mistakenly directed; the correct approach should directly tackle the integral and its relation to \(f(x)\) without overcomplicating the differentiation step.</p>
<p><strong>Step 4:</strong> Revisiting the integral \(\displaystyle\int e^{x^2 + x}(2x^2 + x + 1)\,dx\), consider a substitution that could simplify the expression. Let \(u = x^2 + x\), then \(du = (2x + 1)\,dx\). This substitution does not directly apply because of the \(2x^2\) term. Instead, notice the integral can be broken down into parts that are directly integrable or recognizable in terms of \(e^{x^2 + x}\). The key insight is recognizing the integral as part of a larger derivative, specifically considering how \(e^{x^2 + x}\) relates to its own derivative and the structure of the given integral.</p>
<p><strong>Step 5:</strong> The correct approach involves recognizing the integral as \(\displaystyle\int e^{x^2 + x}(2x^2 + x + 1)\,dx\) can be seen as \(\displaystyle\int e^u \cdot (u + 1)\,du\) if we consider \(u = x^2 + x\), but adjusted for the fact that \(du = (2x + 1)\,dx\), and \(2x^2 + x + 1\) can be seen as \(u + x + 1\), with \(x\) being part of \(du\). However, the direct path to \(f(x)\) involves understanding the given equation as an integral that results in \(e^{x^2} \cdot f(x) + C\), implying \(f(x)\) is derived from integrating \(e^{x^2 + x}(2x^2 + x + 1)\) and then dividing by \(e^{x^2}\) to isolate \(f(x)\). The confusion in steps indicates a need to directly address the integral's structure and its implications for \(f(x)\).</p>
<p><strong>Step 6:</strong> To correctly solve, recognize that \(\displaystyle\int e^{x^2} \cdot e^x(2x^2 + x + 1)\,dx = \displaystyle
Correct Answer: A,B,C,D