Key Concepts and Formulas

• Integration by Parts Formula: $\int u \, dv = uv - \int v \, du$
• Special Integral Form: $\int e^x (f(x) + f'(x)) \, dx = e^x f(x) + C$
• Logarithmic Differentiation: Used to differentiate functions of the form $y = [g(x)]^{h(x)}$. The process involves taking the natural logarithm of both sides: $\ln y = h(x) \ln(g(x))$, and then differentiating implicitly.
• Derivative of $x^x$: The derivative of $x^x$ is $x^x(1 + \ln x)$.

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Step-by-Step Solution

Step 1: Identify the Special Integral Form

The integral is given by $\int\limits_1^2 {{e^x}.{x^x}\left( {2 + {{\log }_e}x} \right)} dx$ The presence of $e^x$ multiplied by another function strongly suggests the use of the special integral form: $\int e^x (f(x) + f'(x)) \, dx = e^x f(x) + C$. Our goal is to manipulate the integrand to fit this pattern.

Step 2: Determine the Candidate for $f(x)$ and its Derivative $f'(x)$

Let's examine the term $x^x \left( {2 + {{\log }_e}x} \right)$. A common strategy when $x^x$ appears is to consider $f(x) = x^x$. We need to find its derivative, $f'(x)$. Using logarithmic differentiation for $y = x^x$: Take the natural logarithm of both sides: $\ln y = \ln(x^x)$ $\ln y = x \ln x$ Differentiate both sides with respect to $x$: $\frac{1}{y} \frac{dy}{dx} = 1 \cdot \ln x + x \cdot \frac{1}{x}$ $\frac{1}{y} \frac{dy}{dx} = \ln x + 1$ Multiply by $y$: $\frac{dy}{dx} = y (\ln x + 1)$ Substitute $y = x^x$: $f'(x) = \frac{d}{dx}(x^x) = x^x (1 + \ln x)$

Step 3: Manipulate the Integrand to Match the Form $e^x (f(x) + f'(x))$

We have $f(x) = x^x$ and $f'(x) = x^x(1 + \ln x)$. Let's see if we can rewrite the integrand $e^x \cdot x^x (2 + \log_e x)$ in the form $e^x (f(x) + f'(x))$. We need to express $x^x(2 + \log_e x)$ as $f(x) + f'(x)$. $f(x) + f'(x) = x^x + x^x(1 + \ln x)$ $f(x) + f'(x) = x^x + x^x + x^x \ln x$ $f(x) + f'(x) = 2x^x + x^x \ln x$ Factor out $x^x$: $f(x) + f'(x) = x^x (2 + \ln x)$ This exactly matches the non-$e^x$ part of our integrand. So, the integral can be written as: $\int\limits_1^2 {{e^x} \left( {x^x + x^x(1 + {{\log }_e}x)} \right)} dx$ This is in the form $\int e^x (f(x) + f'(x)) \, dx$ with $f(x) = x^x$.

Step 4: Apply the Integration Formula and Evaluate the Definite Integral

Using the formula $\int e^x (f(x) + f'(x)) \, dx = e^x f(x) + C$, the indefinite integral is: $\int {{e^x} \left( {x^x + x^x(1 + {{\log }_e}x)} \right)} dx = e^x \cdot x^x + C$ Now, we evaluate the definite integral from $1$ to $2$: $\left[ {{e^x}{x^x}} \right]_1^2$ Substitute the upper limit ($x=2$) and subtract the result of substituting the lower limit ($x=1$): $= \left( e^2 \cdot 2^2 \right) - \left( e^1 \cdot 1^1 \right)$ Calculate each term: For $x=2$: $e^2 \cdot 2^2 = e^2 \cdot 4 = 4e^2$ For $x=1$: $e^1 \cdot 1^1 = e \cdot 1 = e$ Substitute these values back into the expression: $= 4e^2 - e$ Factor out $e$: $= e(4e - 1)$

Step 5: Match the Result with the Options

The calculated value is $e(4e - 1)$. Comparing this with the given options: (A) e(4e + 1) (B) e(2e – 1) (C) e(4e – 1) (D) 4e 2 – 1

Our result matches option (C).

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Common Mistakes & Tips

• Recognizing the Pattern: The key to solving this problem quickly is to identify the $\int e^x(f(x) + f'(x)) \, dx$ pattern. Always look for this when $e^x$ is present.
• Logarithmic Differentiation: Be comfortable with logarithmic differentiation, especially for functions like $x^x$. A mistake here will lead to an incorrect $f'(x)$.
• Algebraic Manipulation: Carefully split and rearrange terms within the parenthesis to correctly identify $f(x)$ and $f'(x)$.
• Definite Integral Calculation: Ensure accurate substitution of the limits of integration, particularly for simple terms like $1^1$.

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Summary

The integral was evaluated by recognizing the special form $\int e^x (f(x) + f'(x)) \, dx$. We identified $f(x) = x^x$ and calculated its derivative $f'(x) = x^x(1 + \ln x)$. The integrand was then rewritten to fit the form $e^x(f(x) + f'(x))$, allowing us to directly apply the integration formula. Evaluating the resulting expression $e^x x^x$ at the limits $2$ and $1$ yielded the final answer.

The final answer is $\boxed{e(4e - 1)}$.