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: $$\mathbf{e^2 + 1}.$$