Question

Evaluate the integral: $\int e^{(e^x + x)} dx$

• A. $e^x + x + c$
• B. $e^{(e^x)} \cdot x + c$
• C. $e^{(e^x)} + c$
• D. $e^{(e^x)}(e^x - 1) + c$

Hint:

Whenever you see a compound exponential function like $e^{(e^x)}$, applying the chain rule to take its derivative immediately gives $\frac{d}{dx}\left(e^{(e^x)}\right) = e^{(e^x)} \cdot e^x = e^{(e^x + x)}$. Because differentiation and integration are inverse operations, noticing this derivative loop instantly proves that option (C) is the right answer without writing out full substitution steps.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem requires finding the indefinite integral of an exponential function where the exponent itself contains another exponential function added to a linear variable.

Step 2: Key Formula or Approach:
Use exponent addition properties to split the integrand: $e^{A+B} = e^A \cdot e^B$. This layout reveals a direct candidate for integration by substitution, matching the classic form: $$\int e^{f(x)} \cdot f'(x) dx = e^{f(x)} + c$$

Step 3: Detailed Explanation:
Let the integral be $I$: $$I = \int e^{(e^x + x)} dx$$ 1. Split the exponential power term into two distinct products using basic law of indices: $$I = \int e^{(e^x)} \cdot e^x dx$$ 2. Apply integration by substitution. Let: $$t = e^x$$ Differentiating both sides with respect to $x$: $$dt = e^x dx$$ 3. Substitute $t$ and $dt$ back into the integral expression: $$I = \int e^t dt$$ Integrating a basic natural exponential function returns itself: $$I = e^t + c$$ 4. Replace the dummy variable $t$ with its original definition $t = e^x$ to complete the problem: $$I = e^{(e^x)} + c$$

Step 4: Final Answer:
The value of the indefinite integral is $e^{(e^x)} + c$, which corresponds to option (C).