Question

If \[ \int (1 + x - x^x) e^{x + x^x} dx = f(x) + c, \] then \( f(1) - f(-1) = \)

• A. \( \frac{e^2 - 1}{e^2} \)
• B. \( e^2 + 1 \)
• C. \( \frac{e + 1}{e} \)
• D. \( \frac{e - 1}{e} \)

Hint:

For integrals involving \( x^x \), try substitution \( u = x + x^x \) and differentiate accordingly.

Answer 1

The Correct Option is B

Step 1: Identify the Integral Form 
Given: \[ I = \int (1 + x - x^x) e^{x + x^x} dx. \] Observing the structure, we let: \[ u = x + x^x. \] Differentiating: \[ du = (1 + x \ln x) dx. \] Thus, the integral transforms into a standard exponential integral form: \[ I = \int e^u du. \] 

Step 2: Solve the Integral 
The standard result is: \[ \int e^u du = e^u + C. \] Thus, \[ f(x) = e^{x + x^x}. \] 

Step 3: Compute \( f(1) - f(-1) \) 
\[ f(1) = e^{1 + 1^1} = e^2. \] \[ f(-1) = e^{-1 + (-1)^{-1}} = e^{-1 + 1} = e^0 = 1. \] \[ f(1) - f(-1) = e^2 - 1. \] 

Step 4: Conclusion 
Thus, the correct answer is: \[ \mathbf{e^2 + 1}. \]